Relativistic Dynamics Of A Charged Sphere Updating The Lorentz Abraham Model

Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, G...

Author: Arthur Yaghjian

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Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany

The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research to serve the following purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected]

Arthur D. Yaghjian

Relativistic Dynamics of a Charged Sphere Updating the Lorentz-Abraham Model

2nd edition

ABC

Author Arthur D. Yaghjian Visiting Scientist Hanscom Air Force Base MA, 01731-5000, USA Email: [email protected]

Arthur D. Yaghjian, Relativistic Dynamics of a Charged Sphere, 2nd ed., Lect. Notes Phys. 686 (Springer, New York 2006), DOI 10.1007/b98846

Library of Congress Control Number: 2005925981 First edition published as Lecture Notes in Physics: Monographs, Vol. 11, 1992. ISSN 0075-8450 ISBN-10 0-387-26021-8 ISBN-13 978-0-387-26021-1 Printed on acid-free paper c 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America 9 8 7 6 5 4 3 2 1 springeronline.com

To LUCRETIA

Foreword

This is a remarkable book. Arthur Yaghjian is by training and profession an electrical engineer; but he has a deep interest in fundamental questions usually reserved for physicists. He has studied the relevant papers of an enormous literature that accumulated for longer than a century. The result is a fresh and novel approach to old problems providing better solutions and contributing to their understanding. Physicists since Lorentz in the late nineteenth century have looked at the equations of motion of a charged object primarily as a description of a fundamental particle, typically the electron. Since the limitations of classical physics due to quantum mechanics have long been known, Yaghjian considers a macroscopic object, a spherical insulator with a surface charge. He thus avoids the pitfalls that have misguided research in the ﬁeld since Dirac’s famous paper of 1938. The ﬁrst edition of this book, published in 1992, was an apt tribute to the centennial of Lorentz’s seminal paper of 1892 in which he ﬁrst proposed the extended model of the electron. In the present second edition, attention is also paid to very recent work on the equation of motion of a classical charged particle. Mathematical approximations for speciﬁc applications are clearly distinguished from the physical validity of their solutions. It is remarkable how these results call for empirical tests yet to be performed at the necessarily extreme conditions and with suﬃciently high accuracy. In these important ways, the present book thus revives interest in the classical dynamics of charged objects.

Syracuse University 2005

Fritz Rohrlich

Preface to the Second Edition

Chapters 1 through 6 and the Appendices in the Second Edition of the book remain the same as in the First Edition except for the correction of a few typographical errors, for the addition and rewording of some sentences, and for the reformatting of some of the equations to make the text and equations read more clearly. A convenient three-vector form of the equation of motion has been added to Chapter 7 that is used in expanded sections of Chapter 7 on hyperbolic and runaway motions, as well as in Chapter 8. Several references and an index have also been added to the Second Edition of the book. The method used in Chapter 8 of the First Edition for eliminating the noncausal pre-acceleration from the equation of motion has been generalized in the Second Edition to eliminate pre-deceleration as well. The generalized method is applied to obtain the causal solution to the equation of motion of a charge accelerating in a uniform electric ﬁeld for a ﬁnite time interval. Alternative derivations of the Landau-Lifshitz approximation to the LorentzAbraham-Dirac equation of motion are also given in Chapter 8 along with Spohn’s elegant solution of this approximate equation for a charge moving in a uniform magnetic ﬁeld. A necessary and suﬃcient condition is found for this Landau-Lifshitz approximation to be an accurate solution to the exact Lorentz-Abraham-Dirac equation of motion. Many of the additions that have been made to the Second Edition of the book have resulted from illuminating discussions with Professor W.E. Baylis of the University of Windsor, Professor Dr. H. Spohn of the Technical University of Munich, and Professor Emeritus F. Rohrlich of Syracuse University. Dr. A. Nachman of the United States Air Force Oﬃce of Scientiﬁc Research supported and encouraged much of the research that led to the Second Edition of the book.

Concord, Massachusetts 2005

Arthur D. Yaghjian

Preface to the First Edition

This re-examination of the classical model of the electron, introduced by H. A. Lorentz 100 years ago, serves as both a review of the subject and as a context for presenting new material. The new material includes the determination and elimination of the basic cause of the pre-acceleration, and the derivation of the binding forces and total stress-momentum-energy tensor for a charged insulator moving with arbitrary velocity. Most of the work presented here was done while on sabbatical leave as a guest professor at the Electromagnetics Institute of the Technical University of Denmark. I am indebted to Professor Jesper E. Hansen and the Danish Research Academy for encouraging the research. I am grateful to Dr. Thorkild B. Hansen for checking a number of the derivations, to Marc G. Cote for helping to prepare the ﬁnal camera-ready copy of the manuscript, and to Jo-Ann M. Ducharme for typing the initial version of the manuscript. The ﬁnal version of the monograph has beneﬁted greatly from the helpful suggestions and thoughtful review of Professor F. Rohrlich of Syracuse University, and the perceptive comments of Professor T. T. Wu of Harvard University.

Concord, Massachusetts April, 1992

Arthur D. Yaghjian

Contents

1

Introduction and Summary of Results . . . . . . . . . . . . . . . . . . . . .

2

Lorentz-Abraham Force and Power Equations . . . . . . . . . . . . . 11 2.1 Force Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Power Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3

Derivation of Force and Power Equations . . . . . . . . . . . . . . . . . . 17 3.1 General Equations of Motion from Proper-Frame Equations . . . 19

4

Internal Binding Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Poincar´e Binding Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Binding Forces at Arbitrary Velocity . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Electric Polarization Producing the Binding Forces . . . .

5

Electromagnetic, Electrostatic, Bare, Measured, and Insulator Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bare Mass in Terms of Electromagnetic and Electrostatic Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Extra Momentum-Energy in Newton’s Second Law of Motion for Charged Particles . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Reason for Lorentz Setting the Bare Mass Zero . . . . . . . .

6

Transformation and Redeﬁnition of Force-Power and Momentum-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Transformation of Electromagnetic, Binding, and Bare-Mass Force-Power and Momentum-Energy . . . . . . . . . . . . . 6.1.1 Total Stress-Momentum-Energy Tensor for the Charged Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Redeﬁnition of Electromagnetic Momentum and Energy . . . . . .

1

23 23 27 32

35 38 40 42

45 45 48 54

XIV

Contents

7

Momentum and Energy Relations . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.1 Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Runaway Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8

Solutions to the Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . 67 8.1 Solution to the Equation of Rectilinear Motion . . . . . . . . . . . . . . 68 8.1.1 Formal Solution to the General Equation of Motion . . . . 72 8.2 Cause and Elimination of the Pre-Acceleration . . . . . . . . . . . . . . 73 8.2.1 Cause of the Pre-Acceleration . . . . . . . . . . . . . . . . . . . . . . . 76 8.2.2 Elimination of the Pre-Acceleration . . . . . . . . . . . . . . . . . . 79 8.2.3 Determination of the Transition Force for Rectilinear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2.4 Motion of Charge in a Uniform Electric Field for a Finite Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2.5 Conservation of Momentum-Energy in the Causal Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.3 Power Series Solution to the Equation of Motion . . . . . . . . . . . . 98 8.3.1 Power Series Solution to Rectilinear Equation of Motion 100 8.3.2 Power Series Solution to General Equation of Motion . . 102 8.3.3 Charge Moving in a Uniform Magnetic Field . . . . . . . . . . 108 8.4 The Finite Diﬀerence Equation of Motion . . . . . . . . . . . . . . . . . . 112 8.5 Renormalization of the Equation of Motion . . . . . . . . . . . . . . . . . 115

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A

Derivation and Transformation of Small-Velocity Force and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1 Derivation of the Small-Velocity Force and Power . . . . . . . . . . . . 121 A.1.1 Derivation of the Proper-Frame Force . . . . . . . . . . . . . . . . 121 A.1.2 Derivation of the Small-Velocity Power . . . . . . . . . . . . . . . 124 A.2 Relativistic Transformation of the Small-Velocity Force and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2.1 Relativistic Transformation of the Proper-Frame Force . 125 A.2.2 Relativistic Transformation of the Small-Velocity Power 126 A.3 Noncovariance of the Power Equation . . . . . . . . . . . . . . . . . . . . . . 127

B

Derivation of Force and Power at Arbitrary Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.1 The 1/a Terms of Self Electromagnetic Force and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.1.1 Evaluation of 1/a Term of Self Electromagnetic Force . . 130 B.1.2 Evaluation of 1/a Term of Self Electromagnetic Power . 133 B.2 Radiation Reaction of Self Electromagnetic Force and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.2.1 Evaluation of the Radiation Reaction Force . . . . . . . . . . . 134

Contents

XV

B.2.2 Evaluation of the Radiation Reaction Power . . . . . . . . . . 137 C

Electric and Magnetic Fields in a Spherical Shell of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

D

Derivation of the Linear Terms for the Self Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

1 Introduction and Summary of Results

The primary purpose of this work is to determine an equation of motion for the classical Lorentz model of the electron that is consistent with causal solutions to the Maxwell-Lorentz equations, the relativistic generalization of Newton’s second law of motion, and Einstein’s mass-energy relation. (The latter two laws of physics were not discovered until after the original works of Lorentz, Abraham, and Poincar´e. The hope of Lorentz and Abraham for deriving the equation of motion of an electron from the self force determined by the Maxwell-Lorentz equations alone was not fully realized.) The work begins by reviewing the contributions of Lorentz, Abraham, Poincar´e, and Schott to this century-old problem of ﬁnding the equation of motion of an extended electron. Their original derivations, which were based on the Maxwell-Lorentz equations and assumed a zero bare mass, are modiﬁed and generalized to obtain a nonzero bare mass and consistent force and power equations of motion. By looking at the Lorentz model of the electron as a charged insulator, general expressions are derived for the binding forces that Poincar´e postulated to hold the charge distribution together. A careful examination of the classic Lorentz-Abraham derivation reveals that the self electromagnetic force must be modiﬁed during a short time interval after the external force is ﬁrst applied and after all other nonanalytic points in time of the external force. The resulting modiﬁcation to the equation of motion, although slight, eliminates the noncausal pre-acceleration (and pre-deceleration) that has plagued the solution to the Lorentz-Abraham equation of motion. As part of the analysis, general momentum and energy relations are derived and interpreted physically for the solutions to the equation of motion, including “hyperbolic” and “runaway” solutions. Also, a stress-momentum-energy tensor that includes the binding, bare-mass, and electromagnetic momentum-energy densities is derived for the charged insulator model of the electron, and an assessment is made of the redeﬁnitions of electromagnetic momentum-energy that have been proposed in the past to obtain a consistent equation of motion. Many ﬁne articles have been written on the classical theories of the electron, such as [7], [32], [41], [42], [52], [71], and [72], to complement the original

2

1 Introduction and Summary of Results

works by Lorentz [4], Abraham [3], Poincar´e [19], and Schott [16]. However, in returning to the original derivations of Lorentz, Abraham, Poincar´e, and Schott, re-examining them in detail, modifying them when necessary, and supplementing them with the results of special relativity not contained explicitly in the Maxwell-Lorentz equations, it is possible to clarify and resolve a number of the subtle problems that have remained with the classical theory of the Lorentz model of the extended electron. An underlying motivation to the present analysis is the idea that one can separate the problem of deriving the equation of motion of the extended model of the electron from the question of whether the model approximates an actual electron. Hypothetically, could not one enter the classical laboratory, distribute a charge e uniformly on the surface of an insulating sphere of radius a, apply an external electromagnetic ﬁeld to the charged insulator and observe a causal motion predictable from the relativistically invariant equations of classical physics? Moreover, the short-range polarization forces binding the excess charge to the surface of the insulator need not be postulated, but should be derivable from the relativistic generalization of Newton’s second law of motion applied to both the charge and insulator, and from the requirement that the charge remain uniformly distributed on the spherical insulator in its proper inertial frame of reference. A summary of the results in each of the succeeding chapters follows. Chapter 2 introduces the original Lorentz-Abraham force and power equations of motion for Lorentz’s relativistically rigid model of the electron moving without rotation1 with arbitrary velocity. Lorentz and Abraham derived their force equation of motion by determining the self electromagnetic force induced by the moving charge distribution upon itself, and setting the sum of the externally applied and self electromagnetic force equal to zero, that is, they assumed a zero “bare mass.” Similarly, they derived their power equation of motion by setting the sum of the externally applied and self electromagnetic power (work done per unit time by the forces on the charge distribution) equal to zero. To the consternation of Abraham and Lorentz, these two equations of motion were not consistent. In particular, the scalar product of the velocity of the charge center with the self electromagnetic force (force equation of motion) did not equal the self electromagnetic power (power equation of motion). Merely introducing a nonzero bare mass into the equations of motion does not remove this inconsistency between the force and power equations of motion. Moreover, it is shown that the apparent inconsistency between self electromagnetic force and power is not a result of the electromagnetic mass in 1

The work of Nodvik [8, eq. (7.28)] shows that the eﬀect of a ﬁnite angular velocity of rotation on the self force and power of the Lorentz model approaches zero to the order of the radius of the charge as it approaches zero and thus classical rotational eﬀects are of the same order as the higher order terms neglected in the Lorentz-Abraham equations of motion.

1 Introduction and Summary of Results

3

the equations of motion equaling 4/3 the electrostatic mass, nor a necessary consequence of the electromagnetic momentum-energy not transforming like a four-vector. The 4/3 factor occurs in both the force and power equations of motion, (2.1) and (2.4), and it was of no concern to Abraham, Lorentz, or Poincar´e in their original works which, as mentioned above, appeared before Einstein proposed the mass-energy relationship. Neither the self electromagnetic force-power nor the momentum-energy transforms as a four-vector. (For this reason, they are referred to herein as force-power and momentum-energy rather than four-force and four-momentum.) However, there are any number of force and power functions that could be added to the electromagnetic momentum and energy that would make the total momentum-energy (call it Gi ) transform like a four-vector, and yet not satisfy dGi /ds ui = 0, so that the inconsistency between the force and power equations of motion would remain. Conversely, it is possible for the proper time derivatives of momentum and energy (force-power) to transform as a four-vector and satisfy dGi /ds ui = 0 without the momentum-energy Gi itself transforming like a four-vector. In fact, Poincar´e introduced binding forces that removed the inconsistency between the force and power equations of motion, and restored the force-power to a four-vector, without aﬀecting the 4/3 factor in these equations or requiring the momentum and energy of the charged sphere to transform as a four-vector. The apparent inconsistency between the self electromagnetic force and power is investigated in detail in Chapter 3 by reviewing the Abraham-Lorentz derivation and rigorously rederiving the electromagnetic force and power for a charge moving with arbitrary velocity. For the Lorentz model of the electron moving with arbitrary velocity, one ﬁnds that the Abraham-Lorentz derivation depends in part on diﬀerentiating with respect to time the velocity in the electromagnetic momentum and energy determined for a charge distribution moving with constant velocity. Although Lorentz and Abraham give a plausible argument for the validity of this procedure, the ﬁrst rigorous derivation of the self electromagnetic force and power for the Lorentz electron moving with arbitrary velocity was given by Schott in 1912, several years after the original derivations of Lorentz and Abraham. Because Schott’s rigorous derivation of the electromagnetic force and power, obtained directly from the Li´enardWiechert potentials for an arbitrarily moving charge, is extremely involved and diﬃcult to repeat, a much simpler, yet rigorous derivation is provided in Appendix B. It is emphasized in Section 3.1 that the self electromagnetic force and power are equal to the internal Lorentz force and power densities integrated over the charge-current distribution of the extended electron, and thus one has no a priori guarantee that they will obey the same relativistic transformations as an external force and power applied to a point mass. An important consequence of the rigorous derivations of the electromagnetic force and power of the extended electron, with arbitrary velocity, is that the integrated self electromagnetic force, and thus the Lorentz-Abraham force equation of motion

4

1 Introduction and Summary of Results

of the extended electron, is shown to transform as an external force applied to a point mass. However, the rigorous derivations also reveal that the integrated self electromagnetic power, and thus the Lorentz-Abraham power equation of motion, for the relativistically rigid model of the extended electron do not transform as the power delivered to a moving point mass. This turns out to be true even when the radius of the charged sphere approaches zero, because the internal ﬁelds become singular as the radius approaches zero and the velocity of the charge distribution is not the same at each point on a moving, relativistically rigid shell. Thus, it is not permissible to use the simple point-mass relativistic transformation of power to ﬁnd the integrated self electromagnetic power of the extended electron in an arbitrarily moving inertial reference frame from its small-velocity value. (This is unfortunate because the proper-frame and small-velocity values of self electromagnetic force and power, respectively, are much easier to derive than their arbitrary-frame values from a series expansion of the Li´enard-Wiechert electric ﬁelds; see Appendix A.) The rigorous derivations of self electromagnetic force and power in Chapter 3 critically conﬁrm the discrepancy between the Lorentz-Abraham force and power equations of motion. Chapter 4 introduces a more detailed picture of the Lorentz model of the electron as a charge uniformly distributed on the surface of a nonrotating insulator that remains spherical with radius a in its proper inertial reference frame. (The values of the permittivity and permeability inside the insulating sphere are assumed to equal those of free space.) Applying the relativistic version of Newton’s second law of motion to the surface charge and insulator separately, we prove the remarkable conclusion of Poincar´e that the discrepancy between the Lorentz-Abraham force and power equations of motion is caused by the neglect of the short-range polarization forces binding the charge to the surface of the insulator. Even though these short-range polarization forces need not contribute to the total self force or rest energy of formation, they add to the total self power an amount that exactly cancels the discrepancy between the Lorentz-Abraham force and power equations of motion. Moreover, the power equation of motion modiﬁed by the addition of the power delivered by the binding forces now transforms relativistically like power delivered to a point mass. With the addition of Poincar´e binding forces, the power equation of motion of the Lorentz model of the electron derives from the Lorentz-Abraham force equation of motion, and no longer needs separate consideration. Of course, Poincar´e did not know what we do today about the nature of these surface forces when he published his results in 1906, so he simply assumed the necessity of “other forces or bonds” that transformed like the electromagnetic forces. Also, Poincar´e drew his conclusions from the analysis of the ﬁelds and forces of a charged sphere moving with constant velocity; see Section 4.1. The derivation in Section 4.2 from the relativistic version of Newton’s second law of motion reveals, in addition to the original Poincar´e stress, both “inhomogeneous” and “homogeneous” surface stresses that are

1 Introduction and Summary of Results

5

required to keep the surface charge bound to the insulator moving with arbitrary center velocity. The extra inhomogeneous stress integrates to zero when calculating the total binding force and power. The extra homogeneous binding force and power just equal the negative of the time rate of change of momentum and energy needed to accelerate the mass of the uncharged insulator. It also vanishes when the mass of the uncharged insulator is zero. The mass of the uncharged insulator should not be confused with the “bare mass” of the surface charge. Today the bare mass should be viewed as simply a mathematically deﬁned mass required to make the Lorentz-Abraham force equation of motion compatible with the relativistic version of Newton’s second law of motion and the Einstein mass-energy relation. Also, the analysis in Section 4.2 conﬁrms the original results of Poincar´e that the forces binding the charge to the insulator remove the inconsistency between the LorentzAbraham force and power equations of motion (that is, between self force and power), but do not remove the 4/3 factor multiplying the electrostatic mass in the equations of motion or require the momentum-energy to transform as a four-vector. With the addition of the binding forces, the force-power, but not the momentum-energy, transforms as a four-vector. Chapter 5 determines the relationships between the various masses (electromagnetic, electrostatic, bare, measured, and insulator masses) involved with the analysis of the classical model of the electron as a charged insulator. Speciﬁcally, the Einstein mass-energy relation demands that the measured mass of the charged insulator equals the sum of the electrostatic mass and the mass of the uncharged insulator (which can include any mass, positive or negative, due to gravitational ﬁelds and the short-range polarization forces binding the charge to the insulator, if their contribution to the rest energy of formation is not negligible). The relativistic version of Newton’s second law of motion then demands that the momentum of the so-called bare mass equals the diﬀerence between the momentum of the electromagnetic mass and the electrostatic mass, regardless of the value of the mass of the insulator. Thus, the ﬁnal analysis shows what one might expect initially, namely, that the self force derived from the Maxwell-Lorentz equations determines the radiation reaction term in the Lorentz-Abraham (or renormalized Lorentz-Abraham-Dirac) equation of motion but not the correct mass in the relativistic Newtonian acceleration term (whether or not the Poincar´e binding forces are included). It is the negative bare mass that removes the 4/3 factor from the electrostatic mass in the Lorentz-Abraham(-Poincar´e) equation of motion and makes the momentum of the charged insulator compatible with the electrostatic rest energy of formation. With the inclusion of both the bare mass and binding stresses, the momentum-energy as well as force-power transform as four-vectors. Why Lorentz, Abraham, and the general physics community assumed as late as 1915 that the bare mass was zero is explained in Sec. 5.1.2. The ﬁnal result of the analysis of Chapter 5 is an equation of motion (5.12) for a charged insulator compatible with the Maxwell-Lorentz equations, the

6

1 Introduction and Summary of Results

relativistic version of Newton’s second law of motion, and the Einstein massenergy relation. (The possibility, considered by Dirac, of extra momentumenergy terms in the relativistic version of Newton’s second law of motion for charged particles, and the conditions these terms should satisfy, are discussed in Section 5.1.1.) Chapter 6 begins by summarizing the transformation properties of the diﬀerent force-powers and momentum-energies, and deriving a total stressmomentum-energy tensor that accounts for the binding forces and bare mass, as well as the electromagnetic self force for the charged insulator model of the electron. We then consider the redeﬁnitions of electromagnetic momentum-energy that have been proposed to obtain consistent momentum and energy equations of motion without introducing speciﬁc binding forces and bare masses. With the exception of the momentum-energy of Schwinger’s tensors [23], the redeﬁned momentum-energy densities can be found for the Lorentz model of the electron by multiplying the four-velocity of the center of the extended charge by an invariant function of the electromagnetic ﬁeld. The total momentum-energy of the charge distribution moving with constant velocity then transforms as a four-vector, and for arbitrary velocity predicts consistent 1/a terms for the self force and self power, that is, consistent 1/a terms in the force and power equations of motion. However, these invariant redeﬁnitions of electromagnetic momentum-energy do not predict the correct radiation reaction terms in the equations of motion. Schwinger’s method [23] consists of writing the force-power density as the divergence of a tensor that depends on the charge-current distribution for charge moving with constant velocity. This charge-current tensor is subtracted from the original electromagnetic stress-momentum-energy tensor, to obtain a divergenceless stress-momentum-energy tensor (when the velocity is constant) and a total momentum-energy that transforms as a four-vector. This method produces the correct radiation reaction terms as well as consistent 1/a terms in the force and power equations of motion for arbitrary velocity. The tensor resulting from this method is ambiguous to within an arbitrary divergenceless tensor. Schwinger concentrates on two tensors which, for a thin shell of charge, are equivalent to the stress-momentum-energy tensor derived for the charged insulator when the value of the mass of the insulator is chosen equal to zero and mes /3, where mes is the electrostatic mass. None of these methods of redeﬁning the electromagnetic momentum-energy require the removal of the 4/3 factor multiplying the electrostatic mass in the original equations of motion. They have the drawback for the Lorentz model of the electron of requiring unknown self force and power (electromagnetic or otherwise) that do not equal the Lorentz force and power. Also, none of the redeﬁned stress-momentum-energy tensors recover the secondary binding forces necessary to hold the accelerating charge to the surface of the insulator. Thus, redeﬁning the electromagnetic momentum-energy seems an unattractive alternative to the deterministic binding forces, bare mass, and

1 Introduction and Summary of Results

7

total stress-momentum-energy tensor derived for the charged-insulator model of the extended electron. In Chapter 7, general expressions for the momentum and energy of the moving charge are derived from the equation of motion. The reversible kinetic momentum-energy, the reversible Schott acceleration momentum-energy, and the irreversible radiation momentum-energy are separated in both three and four-vector notation. After the application of an external force to the charged particle, all the momentum-energy that has been supplied by the external force has been converted entirely to kinetic and radiated momentum-energy. However, while the external force is being applied, the momentum-energy is converted to Schott acceleration momentum-energy, as well as kinetic and radiated momentum-energy. An understanding of the “Schott acceleration momentum-energy” as reactive momentum-energy may be gained by looking at time harmonic motion and comparing the energy of the oscillating charge with the reactive energy of an antenna. It is also conﬁrmed that the conservation of momentum-energy is not violated by a charge in hyperbolic motion (relativistically uniform acceleration), or by the homogeneous runaway solutions to the equation of motion. By writing the three-vector equation of motion in an especially compact form, it is proven that the only possible solution to the equation of motion for relativistically uniform acceleration is rectilinear “hyperbolic motion” of the charge under a constant externally applied force in some inertial reference frame. This is the only externally applied force for which the radiation reaction force is zero and the Lorentz-Abraham-Dirac equation of motion reduces to the relativistic version of Newton’s second law of motion. Chapter 8 begins by solving the equation of motion for the extended charge in rectilinear motion. When one neglects the higher order terms (in radius a) of the equation of motion, one obtains the well-known pre-acceleration solution under the two asymptotic conditions that the acceleration approaches zero in the distant future (when the external force approaches zero in the distant future) and the velocity approaches zero in the remote past. It is shown that this pre-acceleration solution, which violates causality, is not a strictly valid solution to the equation of motion of the extended charge because the pre-acceleration does not satisfy the requirement that the neglected higher order terms in a are negligible. Unfortunately, when higher order terms in the Lorentz-Abraham(-Poincar´e) equation of motion are retained, the noncausal pre-acceleration remains; its time dependence merely changes. In Section 8.2.1 the root cause of the noncausal pre-acceleration solution is traced to the assumption in the classical derivation of the self electromagnetic force that the position, velocity, and acceleration of each element of charge at the retarded time can be expanded in a Taylor series about the present time. With a ﬁnite external force that is zero for all time less than zero and yet an analytic function of time about the real t axis for all time greater than zero, these Taylor series expansions are valid for all time except during

8

1 Introduction and Summary of Results

the initial short time interval light takes to traverse the charge distribution (0 ≤ t ≤ Δta ). It is shown in Section 8.2.2 that when the derivation of the self force is done properly near t = 0, a correction force f a (t) that is nonzero only in the “transition interval” [0, Δta ] must be included in the equation of motion.2 This small correction force in the equation of motion removes the noncausal pre-acceleration from the solution to the equation of motion without destroying the covariance of the equation of motion. In Section 8.2.3 the correction force fa (t) is determined for rectilinear motion in terms of the change in the velocity of the charge across the transition interval. In Section 8.2.4, the corrected equation of motion is applied to the problem of determining the motion of a charge that is accelerated by a uniform electric ﬁeld for a ﬁnite time interval, for example, between the parallel plates of a charged capacitor. With the addition of correction forces fa1 (t) and fa2 (t) at the two nonanalytic points of time in the external force (one when the charge enters the ﬁrst plate and one when it exits the second plate) both pre-acceleration and pre-deceleration are eliminated. Section 8.2.5 reveals that the removal of the noncausal pre-acceleration and pre-deceleration in the equation of motion comes at a cost. Unless the magnitude of the externally applied force is bounded by a ﬁnite (though extremely large) value, no change in velocity across the transition intervals can be chosen to avoid a negative energy radiated during the transition intervals while maintaining causality. For the charged spherical insulator (extended model of the electron), this restriction on the magnitude of the external force is of little concern because it is identical to a proper-frame condition required for neglecting the terms of higher order in the equation of motion than the radiation reaction term. However, for the Lorentz-Abraham-Dirac equation of motion of a point charge (considered in Section 8.5), obtained by letting the radius of the charge approach zero while renormalizing the mass to a ﬁxed ﬁnite value, the higher order terms vanish. Thus, the Lorentz-Abraham-Dirac equation of motion of a mass-renormalized point charge corrected by the transition forces can be made to satisfy causality but at the expense of producing an unphysical negative radiated energy during the transition intervals if the externally applied force becomes extraordinarily large. If one is not concerned with the correct behavior of the solution to the equation of motion during the time immediately after the external force is ﬁrst applied, one can obtain a convenient power series solution to the equa2

It is assumed throughout the book that the fundamental equation of motion for a charged particle is ultimately obtained by equating the sum of the external force and the radiation reaction part of the self force to the rest mass of the particle times the relativistic acceleration. During the transition interval this fundamental equation of motion diﬀers from the equation of motion one would obtain by equating the sum of the external force and the total self force to a constant “bare mass” times the relativistic acceleration. This diﬀerence is important because it allows for a causal (no pre-acceleration or pre-deceleration) initial-value solution to the equation of motion.

1 Introduction and Summary of Results

9

tion of motion. Speciﬁcally, power series solutions and conditions for their convergence are derived in Section 8.3 by the method of successive substitutions for the proper-frame, the rectilinear, and the general equation of motion. The ﬁrst two terms of the power series solution to the general equation of motion are converted to the approximation derived by Landau and Lifshitz [51, sec. 76] to the Lorentz-Abraham-Dirac equation of motion. For the special case of a charge moving in a uniform magnetic ﬁeld, the solution ﬁrst derived by Spohn [38] is given for the Landau-Lifshitz approximation to the LorentzAbraham-Dirac equation of motion. This solution emerges in a simple form convenient for determining the long-term motion of the charge as well as its change in energy and radius of curvature per unit time. The necessary and suﬃcient condition on the magnitude of the applied uniform magnetic ﬁeld is found for this Landau-Lifshitz solution to be an accurate approximation to the exact solution of the Lorentz-Abraham-Dirac equation of motion. Section 8.4 considers the ﬁnite diﬀerence equation of motion of the extended charge that has been proposed as an alternative to the diﬀerential equation of motion. We ﬁnd that there is little justiﬁcation to accept the ﬁnite diﬀerence equation as a valid equation of motion because it neglects all nonlinear terms (in the proper frame of the charge) involving products of the time derivatives of the velocity, and retains a homogeneous runaway solution that leads to pre-acceleration. Section 8.5 concentrates on the mass-renormalized or Lorentz-AbrahamDirac (LAD) equation of motion for a point charge corrected by the transition forces to remove noncausal behavior at nonanalytic points in time of the externally applied force. For a ﬁnite external force that is an analytic function of time about the real t axis for all t except for a ﬁnite number of nonanalytic points in time, the radiation reaction term in the LAD equation of motion is exact for all t except during the inﬁnitesimal transition intervals following the nonanalytic points in time. Thus, the corrected LAD equation of motion merely equates the sum of the radiation reaction and external forces on a point charge outside the transition intervals to the relativistic Newtonian acceleration force, and adds delta-function transition forces during the transition intervals to eliminate the noncausal pre-acceleration and pre-deceleration. Unfortunately, as mentioned above, these causal solutions to the mass-renormalized corrected equation of motion of a point charge can predict an unphysical negative radiated energy during the transition intervals if the magnitude of the external force is large enough. Consequently, and quite startlingly, a classical causal equation of motion of a mass-renormalized point charge that maintains a non-negative radiated energy during the transition intervals for arbitrarily large values of the external force must involve a more complicated joining of the external, radiation reaction, and Newtonian acceleration forces than just their summation. A fully satisfactory classical equation of motion of a point charge does not result from the corrected equation of motion for an extended classical model of a charged particle by simply renormalizing the diverging mass to a ﬁnite value as the radius of the charge is allowed to approach zero.

2 Lorentz-Abraham Force and Power Equations

2.1 Force Equation of Motion Toward the end of the nineteenth century Lorentz modeled the electron (“vibrating charged particle,” as he called it) by a spherical shell of uniform surface charge density and set about the diﬃcult task of deriving the equation of motion of this electron model by determining, from Maxwell’s equations and the Lorentz force law, the retarded self electromagnetic force that the ﬁelds of the accelerating charge distribution exert upon the charge itself [1]. (This initial work of Lorentz in 1892 on a moving charged sphere appeared ﬁve years before J.J. Thomson’s “discovery” of the electron. It is summarized in English by J.Z. Buchwald [2, app. 7].) With the help of Abraham,1 a highly successful theory of the moving electron model was completed by the early 1900’s [3], [4]. Before Einstein’s papers [5], [6] on special relativity appeared in 1905, they had derived the following force equation of motion e2 γ 2 3γ 2 d e2 ¨ + 2 (u · u) ˙ u˙ (γu) − Fext = u 2 3 6π0 ac dt 6π0 c c γ2 3γ 2 ¨ + 2 (u · u) ˙ 2 u + O(a) + 2 u·u (2.1) c c with

− 1 γ = 1 − u2 /c2 2 ,

O(am ) ≡

∞

αn (u)an

n=m

for a “relativistically rigid” spherical shell of total charge e and radius a, moving with arbitrary center velocity, u = u(t), and externally applied force 1

Abraham was the ﬁrst author to obtain the equations of motion in (2.1) and (2.4) for the charge moving with arbitrary velocity. Nonetheless, I refer to them as the Lorentz-Abraham (rather than the Abraham-Lorentz) equations of motion because Lorentz ﬁrst obtained the proper-frame equation of motion corresponding to (2.1) many years earlier in 1892 and was the ﬁrst to propose the relativistically rigid (contracting) model of the electron.

12

2 Lorentz-Abraham Force and Power Equations

Fext (t). The speed of light and permittivity in free space are denoted by c and 0 , respectively. The rationalized mksA international system of units is used throughout, and dots over the velocity denote diﬀerentiation with respect to time. “Relativistically rigid” refers to the particular model of the electron, proposed originally by Lorentz, that remains spherical in its proper (instantaneous rest) frame, and in an arbitrary inertial frame is contracted in the direction of velocity to an oblate spheroid with minor axis equal to 2a/γ. Lorentz, however, used the word “deformable” to refer to this model of the electron. (Even a relativistically rigid ﬁnite body cannot strictly exist because it would transmit motion instantaneously throughout its ﬁnite volume. Nonetheless, one makes the assumption of relativistically “rigid motion” to avoid the possibility of exciting vibrational modes within the extended model of the electron [7, pp. 131-132].) The derivation of the diﬀerential equation of motion (2.1) requires that the externally applied force be an analytic function of time (in a neighborhood of the real time axis) for all time. This is discussed in Chapter 8 when dealing with the problem of pre-acceleration. The inﬁnite summation of order a in the equation of motion (2.1) goes to zero as the radius a approaches zero. For a charged sphere of ﬁnite radius a moving with arbitrary velocity, it is diﬃcult to determine suﬃcient conditions on the velocity and its derivatives or on the externally applied force for these O(a) terms to be negligible. However, the inequalities (8.24) in Section 8.2 give the conditions on the time derivatives of velocity in the proper inertial frame of the charged sphere suﬃcient for neglecting the O(a) terms. Speciﬁcally, in the proper inertial frame it is suﬃcient that 1) the fractional changes in the ﬁrst and higher time derivatives of velocity be small during the time it takes light to travel across the charge distribution, and 2) the velocity changes by a small fraction of the speed of light in this time. Alternatively, the inequalities in (8.90) of Section 8.3 show that the O(a) terms are negligible in the proper inertial frame if 1) the fractional changes in the externally applied force and its ﬁrst and higher time derivatives are small during the time light traverses the charge, and 2) the external force is not large enough to change the velocity by a signiﬁcant fraction of c in this time. In the original analyses of the Lorentz model of the electron, as well as throughout this book, it is assumed that the charged sphere moves “without rotation,” that is, the angular velocity of each point on the sphere is zero in its proper frame of reference. Nodvik [8] has generalized the derivation of the classical equation of motion to include rotation. This generalized equation of motion [8, eq. (7.28)] shows that a ﬁnite angular velocity produces a self electromagnetic force of order a and thus (2.1) remains a valid classical equation of motion if the charged sphere has a ﬁnite angular velocity. (Schott [9] also investigates the motion of a “spinning” sphere, but he left the general expressions for the self force and couple in terms of integrals that discourage a direct comparison with Nodvik’s results.)

2.2 Power Equation of Motion

13

The right-hand side of (2.1) is the negative of the self electromagnetic force Fem determined by Lorentz and Abraham for the moving charge distribution. Thus (2.1) expresses Newton’s second law of motion for the shell of charge when the unknown “bare” mass, or “material” mass as Lorentz called it, in Newton’s second law of motion is set equal to zero. (With the acceptance of special relativity [5] and, in particular, the Einstein mass-energy equivalence relation [6], it is no longer valid to assume, as did Lorentz and Abraham, that the bare mass is independent of the electrostatic energy of formation, that is, independent of the total charge e and radius a. We shall return in Chapter 5 to the subject of the bare mass and the question of why Lorentz et al. believed the bare mass of the electron was negligible.) Remarkably, the special relativistic factor γ in the time rate of change of momentum (ﬁrst term on the right-hand side of (2.1)) and the radiation reaction part of the self force with coeﬃcient e2 /(6π0 c3 ) that doesn’t depend on the size or shape of the charge (second term on the right-hand side of (2.1)) were both correctly revealed, so that (2.1) is invariant to a relativistic transformation from one inertial reference frame to another. That is, both sides of the force equation of motion (2.1) transform covariantly. Moreover, one could choose the radius a such that the inertial electromagnetic rest mass mem =

e2 6π0 ac2

(2.2)

equaled the measured rest mass of the electron. (This value of a equals 4/3 times the “classical radius of the electron.”)

2.2 Power Equation of Motion As long as Lorentz and Abraham limited themselves to the derivation of the force equation of motion (2.1), they saw no inconsistencies in the Lorentz model of the electron. Lorentz was unconcerned with the terms of order a that are neglected in the self force because he assumed the predicted radius of the electron was both realistic and small enough that only the “next term of the series [the radiation reaction term in (2.1)] makes itself felt” [4, sec. 37]. Lorentz and Abraham were also unconcerned with the electromagnetic mass mem in (2.1) equaling 4/3 the electrostatic mass mes , deﬁned as the energy of formation of the spherical charge divided by c2 mes =

e2 8π0 ac2

(2.3)

because they derived the equation of motion (2.1) before Einstein’s 1905 papers on relativistic electrodynamics [5] and the mass-energy relation [6]. In neither of the original editions of their books [3], [4] do they mention the 4/3 factor in the inertial electromagnetic mass of (2.1) being incompatible with

14

2 Lorentz-Abraham Force and Power Equations

the electrostatic energy of formation, or, conversely, the energy of formation of the electron having to equal c2 times the inertial electromagnetic mass.2 In 1904, however, Abraham [10], [3, secs. 15 and 22], [4, sec. 180] derived the following power equation of motion for the Lorentz relativistically rigid model of the electron by determining from Maxwell’s equations the time rate of change of work done by the internal electromagnetic forces (self electromagnetic power)

1 e2 γ 4 3γ 2 e2 d 2 ¨ + 2 (u · u) ˙ Fext · u = γ− − u·u + O(a) (2.4) 6π0 a dt 4γ 6π0 c3 c with O(am ) ≡

∞

αn (u)an .

n=m

As Abraham and Lorentz pointed out, the power equation of motion (2.4) is not consistent with the force equation of motion (2.1). Speciﬁcally, taking the scalar product of the center velocity u with equation (2.1) gives 3γ 2 e2 dγ e2 γ 4 2 ¨ + 2 (u · u) ˙ Fext · u = u·u + O(a) (2.5) − 6π0 a dt 6π0 c3 c which diﬀers from (2.4) by the term e2 d − 24π0 a dt

1 . γ

(2.6)

This is the discrepancy between the force equation of motion and the power equation of motion for the Lorentz model that concerned Abraham and Lorentz, namely, that the scalar product of u with the time rate of change of the electromagnetic momentum did not equal the time rate of change of the work done by the internal electromagnetic forces. Unlike the force equation of motion (2.1), the left- and right-hand sides of the power equation of motion (2.4) do not transform covariantly; see Appendix A. Moreover, neither the force-power on the right-hand sides of (2.1) and (2.4) nor the momentum-energy transforms as a four-vector; see Section 6.1. (Lorentz and Abraham did not mention and were probably not aware of this noncovariance because these equations were discussed outside the general framework and without the correct velocity transformations of special relativity; compare [11] with [5].) After the derivation of (2.4), they still saw no problem with the 4/3 factor in the inertial electromagnetic mass, nor with the conventional electromagnetic momentum-energy per se (before taking the time derivative) failing to 2

The second edition (1908) of Abraham’s book added to the ﬁrst edition [3] a discussion of the theory of relativity and a section 49 in which he mentions the 4/3 factor.

2.2 Power Equation of Motion

15

transform as a relativistic four-vector. Moreover, if one rewrites the LorentzAbraham equation of motion (2.1) in four-vector notation

2 i j e2 d u e2 dui i i duj du − + O(a) (2.7) Fext = + u 6π0 a ds 6π0 ds2 ds ds one recovers equation (2.1) and (2.5) from (2.7) and misses entirely the discrepancy introduced by the power equation of motion (2.4) derived from Maxwell’s equations by Abraham. If the mass in (2.7) is “renormalized” to a ﬁnite value as the radius of the charge approaches zero and the O(a) terms vanish, then (2.7) becomes identical to the Lorentz-Abraham-Dirac equation of motion [12], [13]; see Section 8.5. Apparently, the radiation reaction, [e2 /(6π0 )][d2 ui /ds2 + ui (duj /ds)(duj /ds)], in the equation of motion was ﬁrst written in four-vector form by von Laue [14]. Early use of the fourvector notation for the radiation reaction in the equation of motion (2.7) can be found in Pauli’s article on relativity theory [7, sec. 32]. Herein we use the four-vector notation of Panofsky and Phillips [13], who normalized the four-velocity to be dimensionless. Throughout this book the entire von Laue term is referred to as the “radiation reaction” rather than just the ui (duj /ds)(duj /ds) part of this term. That is, the “Schott term” (d2 ui /ds2 term) is considered part of the radiation reaction. Only then does the “radiation reaction” in an arbitrary inertial ¨ radiation reaction term (see (3.3)) derived by frame reduce to the original u Lorentz in the proper (u = 0) inertial frame of the charged sphere.

3 Derivation of Force and Power Equations

The inconsistency between the power and force equations of motion, (2.4) and (2.1) or (2.5), is so surprising that one is tempted to question the LorentzAbraham derivation ([10], [3, secs. 15 and 22], [4, sec. 180]) of (2.1) and (2.4). Thus, let us take a careful look at their method of derivation. The right-hand side of (2.1) is the negative of the self electromagnetic force, Fem , and the right-hand side of (2.4) is the negative of the work done per unit time, Pem , by the internal electromagnetic forces (self electromagnetic power) on the moving shell of charge; speciﬁcally d Fem (t) = ρ(r, t)[E(r, t) + u(r, t) × B(r, t)]dV = − 0 E × B dV (3.1) dt charge

all space

Pem (t) =

ρ(r, t)u(r, t) · E(r, t)dV = −

charge

d 0 dt 2

(E 2 + c2 B 2 )dV

(3.2)

all space

where ρ(r, t) and u(r, t) are the density and velocity of the charge distribution in the shell, and E(r, t) and B(r, t) are the electric and magnetic ﬁelds produced by this moving charge distribution. The magnetic ﬁeld does not appear in the ﬁrst integral of (3.2) because the magnetic force is perpendicular to the velocity. (Some authors refer to the radiation reaction term alone in (2.1) or (2.4) as the self force or self power, respectively. However, this seems undesirable because (3.1) and (3.2) clearly deﬁne the Lorentz self electromagnetic force and power, and they are equal to the negative of the right-hand side of (2.1) and (2.4), respectively.) The second equations in (3.1) and (3.2) are, of course, identities derived from Maxwell’s equations, assuming there are no radiation ﬁelds beyond a ﬁnite distance from the charge distribution [15, sec. 2.5, eq.(25) and sec. 2.19, eq.(6)]. For the Lorentz relativistically rigid model of the electron, the charge density and velocity of each part of the shell cannot be the same for an arbitrarily moving shell if the shell is to maintain its spherical shape and uniform charge

18

3 Derivation of Force and Power Equations

density in its proper frame of reference (inertial frame at rest instantaneously with respect to the center of the electron). In particular, the relativistic contraction of the moving Lorentz model of the electron, from a spherical to an oblate spheroidal shell, demands that the velocity of its charge distribution cannot be uniformly equal (except in the proper frame) to the velocity of the center of the shell denoted simply by u = u(t) in our previous equations (see Appendix A). If u(r, t) did not depend on the position r within the shell, as in Abraham’s noncontracting (nonrelativistically rigid) model of the electron [3], u(r, t) could be brought outside the charge integrals in (3.1) and (3.2), Pem would equal Fem · u, and the discrepancy (2.6) between (2.4) and (2.5) or (2.1) would vanish. Such a model is unrealistic because it would have a single preferred inertial frame of reference in which it were spherical (with ﬁxed radius a) and its major axis would stretch to an inﬁnite length in its proper frame when its velocity with respect to the preferred frame approached the speed of light. Still we can ask if the variable velocity in the charge integrals of (3.1) and (3.2) for the Lorentz model of the electron actually produces the discrepancy (2.6) between equations (2.4) and (2.5) or (2.1). For a charge with velocity other than zero, both Abraham and Lorentz derived the ﬁrst terms on the right-hand sides of (2.1) and (2.4), the terms in question, not from the charge integrals in (3.1) and (3.2) but by evaluating the momentum and energy integrals (second integrals) in (3.1) and (3.2) for a charge moving with constant velocity with respect to time, then diﬀerentiating the resulting functions of velocity with respect to time [3, sec. 22], [4, sec. 180]. We know that falsely setting the charge velocity u(r, t) independent of r in the ﬁrst integrals of (3.1) and (3.2) eliminates the discrepancy (2.6). Therefore, is it really justiﬁable, as Lorentz [4, sec. 183] and Abraham [3, sec. 23] argue, to assume a charge velocity constant in time in the second integrals of (3.1) and (3.2) to derive the ﬁrst terms of (2.1) and (2.4), the terms that produce the discrepancy (2.6)? Apparently, this question was not decided with certainty until the work of Schott [16] who derived both the force and power equations of motion, (2.1) and (2.4), by evaluating directly the integrals in (3.1) and (3.2) over the charge distribution for the Lorentz model of the electron moving (without rotation) with arbitrary center velocity u . In particular, his evaluation of the charge integral in (3.2) indeed yielded the power equation of motion (2.4) to prove that the discrepancy (2.6) with the force equation of motion (2.1) actually existed. In fact, Schott’s book appears to be the ﬁrst reference in which either the force or power equation of motion can be found in the general form of (2.1) and (2.4). To obtain these equations from the work of Lorentz and Abraham, one has to piece together the results of a number of their papers or various sections of their books (e.g., secs. 28, 32, 37, 179, and 180 of [4] plus secs. 15 and 22 of [3]). Schott’s derivations of the force and power equations of motion, (2.1) and (2.4), from the charge integrals of (3.1) and (3.2) involve extremely tedious manipulations of the double integrations of the Li´enard-Wiechert potentials for

3.1 General Equations of Motion from Proper-Frame Equations

19

an arbitrarily moving charge distribution. They are so involved that Schott’s rigorous approach to the analysis of the Lorentz model of the electron has not appeared or been repeated, as far as I am aware, in any subsequent review or textbook. Page [17] also derives the force equation of motion (2.1) by evaluating and integrating directly the self electromagnetic ﬁelds over the charge distribution. However, Page’s derivation does not show explicitly the variation in velocity of the charge distribution throughout the shell, and thus it cannot be used to derive the power equation of motion (2.4).

3.1 General Equations of Motion from Proper-Frame Equations Lorentz also derived the force equation of motion from the charge integral for electromagnetic force in (3.1) by means of a double integral of the Li´enardWiechert potentials, but only in the proper frame of the electron where the velocity of the charge is zero and the derivation simpliﬁes greatly to yield the well-known result [4], [13] (derived in Appendix A) Fext =

e2 e2 ¨ + O(a), u˙ − u 2 6π0 ac 6π0 c3

u=0

(3.3)

to which the general force equation of motion reduces when the velocity u in (2.1) is set equal to zero or when (u/c)2 1. (Equation (3.3) was ﬁrst derived in 1892 [1] for a charged sphere even though the electron had not been oﬃcially “discovered” by J.J. Thomson until 1897.) For a velocity much less than the speed of light, a derivation performed in Appendix A, similar to Lorentz’s derivation of (3.3), but applied to the charge integral for electromagnetic power in (3.2), yields the small-velocity power equation of motion Fext · u =

5e2 e2 ˙ ¨ + O(a), u · u − u·u 24π0 ac2 6π0 c3

(u/c)2 1

(3.4)

to which the general power equation of motion (2.4) reduces when only the ﬁrst order terms in u/c are retained. Note once again that the scalar product of u with the force equation (3.3) does not yield the power equation (3.4). Section A.1.2 of Appendix A shows explicitly that the variation of the velocity over the charge distribution, even for (u/c)2 1, must be taken into account to derive the correct expression (3.4) for the small-velocity electromagnetic power. Now equations (3.3) and (3.4) raise an important question. Since the force and power equations of motion, (3.3) and (3.4), are derived rigorously from (3.1) and (3.2) for u approaching zero, why not simply apply the relativistic transformation to the velocity, its time derivatives, and the external force in (3.3) and (3.4) to obtain the general equations of motion (2.1) and (2.4) in an

20

3 Derivation of Force and Power Equations

arbitrary frame. Thereby, one would avoid the diﬃcult evaluation of the self force and power directly from (3.1) and (3.2) for a relativistically rigid shell of charge moving with arbitrary center velocity u . ˙ u ¨ , and Fext in the proper-frame Indeed a relativistic transformation of u, force equation of motion (3.3) produces the general force equation of motion (2.1) [14], [18], [7]. However, the same relativistic transformations applied to (3.4) produce the equation (see Appendix A) e2 γ 4 3γ 2 5e2 dγ 2 ¨ ˙ Fext · u = − u · u + + O(a) (3.5) (u · u) 24π0 a dt 6π0 c3 c2 which does not agree with either the general power equation of motion (2.4) or the equation (2.5) obtained from the scalar product of u with the force equation of motion (2.1). This apparent paradox is explained by returning to (3.1) and (3.2). Since the self force Fem and self power Pem in (3.1) and (3.2) are quantities obtained by integrating over a ﬁnite distribution of charge and are not the force and power applied to a point mass, it is not valid to apply the point relativistic transformations of force and center velocity (and derivatives of velocity) to determine the general values of the integrals in (3.1) and (3.2) from their proper-frame or small-velocity values. For the force equation of motion, the integrated self force (3.1) maintains the transformation properties of a point force, and thus the point relativistic transformations can still be applied to obtain the general integrated self force (3.1) in an arbitrary inertial frame from its proper-frame value on the right-hand side of (3.3). (Unfortunately, one proves this fact by performing the diﬃcult evaluation of (3.1) in the arbitrary inertial frame.) For the power equation of motion, however, the integrated power (3.2) does not transform as the time rate of change of energy of a moving point mass (see Appendix A), even as the radius of the charged shell approaches zero, and thus the point relativistic transformations applied to the small-velocity power (right-hand side of (3.4) as u → 0) do not give the correct value of the power in an arbitrary inertial frame (right-hand side of (2.4)). From the viewpoint of the electromagnetic stress-momentum-energy tensor (discussed in Chapter 6), it is not surprising that the power equation of motion does not transform covariantly, because the electromagnetic stressmomentum-energy tensor of a charged shell is not divergenceless and the electromagnetic momentum-energy does not transform as a four-vector. In summary, then, since the point relativistic transformations do not necessarily apply to an integrated force or power (and the electromagnetic stressmomentum-energy tensor is not divergenceless), it is not mathematically rigorous to use these transformations to ﬁnd the integrated self force and power, (3.1) and (3.2), in an arbitrarily moving inertial reference frame from their proper-frame or small-velocity expressions (3.3) and (3.4). Moreover, as explained above, the classic Lorentz-Abraham derivation of (2.1) and (2.4) for arbitrary u also lacks rigor because it depends upon the evaluation of the

3.1 General Equations of Motion from Proper-Frame Equations

21

momentum and energy of a shell of charge moving with constant, rather than arbitrary time varying velocity. Thus, it appears that Schott’s book [16] contains the only rigorous derivation to date of both the force equation of motion (2.1) and the power equation of motion (2.4). Since this highly commendable derivation by Schott is also extremely tedious and diﬃcult to repeat or check, a much shorter, simpler, yet rigorous derivation of the self electromagnetic force and power is given in Appendix B by applying the relativistic transformations of the electromagnetic ﬁelds at each point within the arbitrarily moving shell of charge before performing the integrations in (3.1) and (3.2). (All these derivations depend upon expanding the position, velocity, and acceleration of each element of the charge at the retarded time in a power series about the present time. When the external force is applied at t = 0, having been zero for t < 0, these series expansions are invalid between t = 0 and t = Δta , the short time interval it takes light to traverse the charge distribution. In Section 8.2 it is shown that the addition of a transition correction force during this short time interval [0, Δta ] eliminates the noncausal pre-acceleration from the solution to the uncorrected equation of motion.)

4 Internal Binding Forces

In Appendix B, we have critically conﬁrmed the evaluation of the self electromagnetic force and power, (3.1) and (3.2), leading to the force and power equations of motion (2.1) and (2.4). Yet (2.1) and (2.4) are inconsistent, since taking the scalar product of u with (2.1) gives (2.5), which diﬀers from (2.4) by the term (2.6). Not only the self electromagnetic momentum-energy but also the self electromagnetic force-power fails to transform as a four-vector. What has gone wrong? To see clearly the problem and its resolution, it helps to divorce the analysis of the moving spherical shell of charge from the question of whether it models the electron. The analysis is based entirely upon classical ﬁelds, forces, and charges, and the extent to which it describes the internal structure of the electron is irrelevant to the question of the inconsistency between the force equation of motion (2.1) and the power equation of motion (2.4). We could enter our classical laboratory, distribute a charge uniformly on the surface of an arbitrarily small, massless (or nearly massless), “relativistically rigid,” insulating sphere, accelerate this charged sphere, and, assumably, get consistent results between the force that is required to accelerate the sphere and the power delivered to the sphere.

4.1 Poincar´ e Binding Forces Poincar´e visualized such a model in his 1906 paper on the dynamics of the electron [19]. (Actually, Poincar´e [19, sec. 6] mentions the charge distributed on a conductor rather than an insulator. We choose the insulator model to avoid the possibility of the charge redistributing itself when the sphere moves. Also, we assume that the values of the relative permittivity and permeability within the spherical insulator are equal to unity so that there is free-space propagation of electromagnetic waves inside as well as outside the sphere.) He argued that the only way the charge could remain on the sphere was for there to exist binding forces exerted on the charge by the insulator that would

24

4 Internal Binding Forces

exactly cancel the repulsive portion of the electromagnetic forces. These internal binding forces are not optional, they are necessary in a stable classical Lorentz model. They are the short-range polarization1 forces that must exist at the surface of the insulator to hold the excess charge to the surface. Although Poincar´e did not have today’s knowledge of the nature of the internal binding forces, he assumed they existed. To quote the English translation of Poincar´e, “Therefore it is indeed necessary to assume [in the Lorentz model] that in addition to electromagnetic forces [of the excess charge alone], there are other forces or bonds” [19, sec. 1]. Thus the total force exerted on the charge in both the force and power equations of motion, (2.1) and (2.4), must include these internal binding forces (which, for the insulator model, are also electromagnetic in origin) as well as the internal electromagnetic forces of the excess charge. For a stationary charged sphere, as Poincar´e explained, the binding forces exerted by the relativistically rigid insulator on the excess charge must be equal and opposite the repulsive electromagnetic forces produced by the excess charge distribution. However, in order to include the binding forces in the force and power equations of motion, one has to know the value of the binding forces for an arbitrarily moving shell of charge. Poincar´e determined the internal binding forces on a moving shell by assuming a “postulate of relativity”, namely that the “impossibility of experimentally demonstrating the absolute movement of the earth would be a general law of nature”; and, in particular, hypothesized with Lorentz [11, sec. 8] that the internal forces in the Lorentz model would obey the same transformations that Maxwell’s equations implied for the electromagnetic forces [19, Introduction]. (Poincar´e did not have the beneﬁt of Einstein’s relativity papers [5], [6] when he submitted his paper [19] in July 1905, or the knowledge that the binding forces could be short-range polarization forces of electromagnetic origin.) As a consequence of this latter hypothesis, Poincar´e drew a startling conclusion. The internal binding forces that canceled the internal self electrostatic forces of the excess charge on the sphere at rest, when transformed to a moving shell, would not contribute to the total self force on the moving charge but would contribute to the total time rate of change of energy (power) delivered to the charge in the Lorentz model of the moving charge. Speciﬁcally, when Poincar´e assumed with Lorentz that the spherical shell compressed to the shape of an oblate spheroid in the direction of its velocity by a factor of 1 − u2 /c2 , the time rate of change of the binding self energy just canceled the discrepancy (2.6) in the power equation of motion (2.4). To see how Poincar´e arrived at this remarkable result, begin with the electrostatic force per unit surface charge 1

The surface forces that bind the excess charge to the surface of the spherical insulator can be regarded classically as resulting from electric polarization induced at the surface of the insulator material; see Section 4.2.1.

4.1 Poincar´e Binding Forces 0 fem =

e ˆr 8π0 a2

25

(4.1)

for a stationary sphere of radius a and total charge e. The binding force per unit charge required to hold the charge on the stationary sphere is then given 0 by the negative of fem or e ˆr . (4.2) fb0 = − 8π0 a2 Now let the charged sphere move with a constant velocity u and contract in the direction of u to an oblate spheroid with minor axis equal to a 1 − u2 /c2 = a/γ. The Lorentz force law and Maxwell’s equations applied to this moving oblate spheroid predict that the electrostatic force per unit charge in (4.1) and thus the binding force per unit charge in (4.2) transforms to 0 0 fb = fb + fb⊥ /γ

(4.3)

where the subscripts, and ⊥, refer to the three-vector components parallel and perpendicular to the velocity u. The transformed binding force in (4.3) is directed along the normal into the surface of the oblate spheroid. The binding force per unit charge (4.3) integrated over the surface charge of the oblate spheroid, because of its symmetry, gives a total binding force Fb equal to zero as in the case of the stationary sphere, that is 0 0 fb + fb⊥ /γ de = 0 . (4.4) Fb = fb de = charge

charge

However, the work taken by the binding forces from the charge distribution as the charge accelerates from zero to velocity u, if we can assume (4.3) is valid for the accelerating charge as well as the charge moving with constant velocity, would be ⎡ ⎤ ⎡ ⎤ a(cos a(cos θ)/γ θ)/γ ⎢ ⎥ ⎢ ⎥ (4.5) fb · dr ⎦ de = − fb0 · dr ⎦ de Wb = − ⎣ ⎣ charge

a cos θ

charge

a cos θ

where θ is the angle between the position vector r to the element of charge de and the velocity u. The charge element de can be expressed as the product of the surface charge density on the sphere (e/(4πa2 )) and the projection of the surface area element of the sphere onto the plane perpendicular to u de =

e dA⊥ . 4πa2 cos θ

(4.6)

From (4.2), the integrand of (4.5) can be rewritten as fb0 · dr = −

e cos θ dr . 8π0 a2

(4.7)

26

4 Internal Binding Forces

Substitution of (4.6) and (4.7) into (4.5) gives ⎡ ⎤ a(cos θ)/γ sphere e2 e2 ⎢ ⎥ Wb = dA dr = − dV ⎣ ⎦ ⊥ 32π 2 0 a4 32π 2 0 a4 sphere

or

a cos θ

e2 Wb = 24π0 a

(4.8)

spheroid

1 −1 γ

.

(4.9)

Equations (4.8) and (4.9) reveal that the work taken by the internal binding forces as the spherical charge distribution accelerates and contracts to the shape of an oblate spheroid is the same as the work taken by a constant pressure, e2 /(32π 2 0 a4 ), on a sphere that is compressed to an oblate spheroid. In the words of the English translation of Poincar´e, “I have attempted to determine this force, and I found that it can be compared to a constant external pressure acting on the deformable and compressible electron, the work of which is proportional to the variations of the volume of this electron” [19, Introduction]. The negative of the time derivative of (4.9) determines the work done per unit time, Pb , by the internal binding forces on the moving charge

d 1 e2 (4.10) Pb = − 24π0 a dt γ that must be subtracted from the right-hand side of the power equation of motion (2.4). Comparing (4.10) with (2.6), we see, as Poincar´e did, that the time rate of change of the work done on the charge by the binding force required to keep the charge on the insulator just cancels the discrepancy (2.6) in power between the power equation of motion (2.4) and the force equation of motion (2.1). As (4.4) shows, the Poincar´e binding forces do not alter, however, the total force on the charge distribution, and thus the force equation of motion (2.1), including the 4/3 factor multiplying the electrostatic mass (2.3), remains unaﬀected by the Poincar´e binding forces. Neither does the power (4.10) delivered by the Poincar´e binding forces remove the 4/3 factor from the power equation of motion (2.4), nor do these binding forces change the rest energy of the charged sphere because Wb in (4.9) vanishes when u is zero. Of course, when the charge is ﬁrst placed on the insulator, the short-range attractive polarization forces holding the charge to the insulator may contribute a negative work of formation to the charge that can subtract from the total rest mass of the charge. The negative mass contributed by these short-range polarization forces binding the charge to the insulator and by other possible attractive forces such as gravity or the short-range forces holding the insulator material together will be included as part of the uncharged insulator mass mins introduced in the next section.

4.2 Binding Forces at Arbitrary Velocity

27

4.2 Binding Forces at Arbitrary Velocity The formulation and integrations of the Poincar´e binding forces in the previous section are based on the ﬁelds and forces of charges in uniform motion. It is uncertain that these results obtained assuming a constant velocity are valid for a shell of charge moving with arbitrary velocity, especially when taking the time derivative of (4.9) to determine the contribution (4.10) of the internal binding forces to the power equation of motion. Thus, we shall derive the polarization binding forces needed to keep the charge on an insulator moving with arbitrary velocity, assuming that the charge remains uniformly distributed on the spherical insulator in its proper inertial frame of reference. (Incidentally, the question raised by Abraham and Lorentz [4, sec. 182] of what keeps the electron in stable equilibrium can be answered for the charged insulator model as the nonclassical energy conﬁgurations keeping the insulating material and excess charge “rigid” in the proper reference frame; see Section 4.2.1.) Consider the shell of total charge e in its proper frame as a uniform distribution of volume charge density located between the radius a and a+δ, where δ is the inﬁnitesimally thin thickness of the spherical shell (see Fig. 4.1). At

Fig.

4.1.

Lorentz model of the electron viewed in its proper frame ˙ ¨ (r, t) . . . . = 0]. [u(r, t) = 0, u(r, t), u

the one instant of time t in its proper frame, the velocity u(r, t) of the charge ˙ t) and at every position r within the shell is zero, but the acceleration u(r, higher time derivatives of velocity are not necessarily zero nor independent of position r within the shell. In Appendix C we determine the internal electric and magnetic ﬁelds in the proper frame of the accelerating shell of charge, and, in particular, ﬁnd the self electromagnetic force per unit charge within the shell to equal

28

4 Internal Binding Forces

fem (r, t) =

(r − a) e 2¨ u 2u˙ ˆr − + 3 2 2 4π0 δa 3ac 3c

¯ 4 I 2 ˙ + O(a), + 4 ˆr · u˙ u˙ − |u| 5c 3

u=0.

(4.11)

(In (4.11), as throughout, when u and its time derivatives are written without the explicit functional dependence (r, t), they refer to the velocity and its time derivatives of the center of the shell.) The force on any volume element of charge in the shell is the sum of the externally applied force, the internal electromagnetic force, and the internal binding force on that element. From Newton’s second law of motion, we assume the sum of these three forces equals an unknown “bare” mass of that charge element multiplied by the acceleration (see Section 5.1). Speciﬁcally fext (r, t) + fem (r, t) + fb (r, t) =

M0 ˙ t), u(r, e

u=0

(4.12)

where fext (r, t), fem (r, t), and fb (r, t) are the external, internal self electromagnetic, and internal binding forces per unit charge, respectively, at the position r in the shell at the instant of time t in the proper frame (u(r, t) = u(t) = 0). The so-called bare mass M0 , which Lorentz set equal to zero, should not be associated with the uncharged mass of the insulator on which the charge is placed. In principle, the mass of the insulator can be made negligible, but M0 on the right-hand side of (4.12) is dependent upon the charge despite its traditional label as “bare” mass. The following derivation shows that the binding force is independent of the value of the bare mass M0 . (The determination of the mass M0 and the reason Lorentz thought it was negligible are discussed in Section 5.1.) In (4.12) we assume the bare mass M0 of the charge is uniformly distributed with the charge in its proper frame so that the bare mass per unit charge at each point in the spherical shell is M0 /e. Similarly, we shall assume that the variation of the external force is negligible over the charge distribution so that it is applied uniformly (to order a) throughout the proper-frame shell, that is Fext (t) fext (r, t) = + O(a) . (4.13) e As a consequence of the shell remaining spherical in its proper inertial frame of reference, we have from equation (A.8) of Appendix A that the acceleration ˙ t) of the charge element at r is related to the acceleration, u˙ = u(t), ˙ u(r, of the center of the nonrotating shell by the formula ˙ t) = u˙ − u(r,

a ˙ u˙ + O(a2 ) . (ˆr · u) c2

(4.14)

Inserting the external force (4.13), the internal self electromagnetic force (4.11), and the acceleration from (4.14) into the equation (4.12), we obtain

4.2 Binding Forces at Arbitrary Velocity

e e M0 ¨ + fb (r, t) u˙ + + u 2 6π0 ac e 6π0 c3

¯I e e(r − a) 2 ˙ ˆ ˆ ˙ ˙ | u| + O(a), r − r · u u − =− 4π0 δa2 5π0 c4 3

Fext − e

29

u = 0.

(4.15)

Next, integrate (4.15) over the entire charge on the shell to get

e2 e2 ˙ ¨ + fb (r, t)de+O(a) = 0, u = 0 (4.16) Fext − + M0 u+ u 6π0 ac2 6π0 c3 charge

since the integral of ˆr over the uniform spherical charge distribution is zero. Divide (4.16) by the total charge e and subtract the result from (4.15) to show that the binding force has to satisfy the equation 1 fb (r, t) − fb (r, t)de e charge

¯I e(r − a) e 2 ˙ ˆ ˆ ˙ ˙ =− | u| + O(a), u = 0. r − r · u u − 4π0 δa2 5π0 c4 3

(4.17)

The most general solution to (4.17) can be found by letting the binding force equal the right-hand side of (4.17) plus an homogeneous solution fbh (r, t)

¯I e(r − a) e 2 ˙ ˆr − ˆr · u˙ u˙ − |u| fb (r, t) = − 4π0 δa2 5π0 c4 3 + fbh (r, t) + O(a), u = 0 . (4.18) Substituting fb (r, t) from (4.18) into (4.17) and again noting that the integral of ˆr over the charge distribution is zero, one sees that the homogeneous solution must satisfy the condition 1 fbh (r, t) = fbh (r, t)de . (4.19) e charge

The right-hand side of (4.19) is not a function of position r, so the left-hand side, fbh (r, t), cannot be a function of r, that is fbh (r, t) = fbh (t)

(4.20)

and (4.19) reduces to an identity. Since we have proven that the homogeneous solution fbh for the binding force is independent of the position of the charge element within the shell, it does not average to zero when integrated over the charge unless it is identically zero. This homogeneous binding force is exerted on the insulator in the opposite direction. Speciﬁcally, if the rest mass of the uncharged insulator is mins (assumed uniformly distributed over the sphere), fbh is given simply as

30

4 Internal Binding Forces

fbh (t) = −

mins u˙ e

(4.21)

from Newton’s second law of motion applied to the insulator in its proper frame. (The inhomogeneous binding force in (4.18) is also exerted in the opposite direction on the insulator but because its total integrated value is zero it does not contribute to the acceleration of the rigid insulator.) With the addition of the homogeneous binding force (4.21), the binding force (4.18) per unit charge needed to keep the charge on the moving insulator is given by

¯I e(r − a) e 2 ˙ ˆr − ˆr · u˙ u˙ − |u| fb (r, t) = − 4π0 δa2 5π0 c4 3 mins − u˙ + O(a), u = 0 . (4.22) e Equation (4.22) shows that the binding force is independent of the bare mass M0 . The ﬁrst term on the right-hand side of (4.22), when integrated over the thickness of the shell of charge, produces the static binding force (4.2) per unit charge given by Poincar´e [19]. The second term on the right-hand side of (4.22) is a binding force that does not appear in Poincar´e’s analysis using a charged shell moving with constant velocity. It is required to cancel the self electromagnetic acceleration forces in (4.11) that vary with position r about the shell. The third term on the right-hand side of (4.22) accounts for the force exerted on the charge to accelerate the mass of the uncharged insulator. If the short-range polarization forces binding the charge to the insulator, or gravitational ﬁelds [20], [21], or other attractive forces such as those that hold the insulator material together contribute to the rest energy of formation, this mass can be included in the mass mins of the uncharged insulator. When we integrate the force per unit charge in (4.22) over the shell, the ﬁrst two terms on the right-hand side of (4.22) vanish to give a total binding force equal to the homogeneous binding force Fb = fb (r, t)de = mins u˙ + O(a), u = 0 (4.23) charge

needed to accelerate the insulator in the proper frame. Furthermore, because the ﬁrst two terms of the internal binding force (4.22) at every point within the charge shell equal the negative of the internal electromagnetic force (4.11), except for the terms in (4.11) that are independent of ˆr, the analyses in Appendices A and B can also be applied to these internal binding forces to obtain the total binding force and the total power delivered to the charge by the binding forces in an arbitrary frame of reference. In particular, the generalization of the second term on the right-hand side of (4.22) to an arbitrary inertial frame integrates to zero when ﬁnding the total binding force, and leads to a term of order a when ﬁnding the total power delivered to the charge by the binding

4.2 Binding Forces at Arbitrary Velocity

31

forces. The ﬁrst term on the right-hand side of (4.22) also integrates to zero in an arbitrary inertial frame but contributes to the power delivered to the charge by the amount given in (2.6) or (4.10) when multiplied by the velocity u(r, t) and integrated over the charge. And, of course, the third term in (4.22) generalizes immediately to −mins d(γu)/dt, which contributes −mins c2 dγ/dt to the power delivered to the charge. Thus, the total binding force and power, contributed by the rigorously derived internal binding force per unit charge needed to keep the charge on an insulator moving with arbitrary velocity, are identical to those given in (4.4) and (4.10) by Poincar´e (for a massless insulator, mins = 0) using binding forces inferred from the ﬁelds and forces of a charge distribution moving with constant velocity, that is d(γu) + O(a), u = 0 (4.24a) fb (r, t)de = −mins Fb (t) = dt charge

Pb (t) =

d e2 fb (r, t)·u(r, t)de = − 24π0 a dt

1 dγ −mins c2 +O(a). (4.24b) γ dt

charge

Recall that the velocity u(r, t) for each portion of the charge distribution cannot equal the velocity u(t) of the center of the shell (except when u(t) = 0) if the shell is to remain spherical in its proper frame of reference (see Appendix A). Thus u(r, t) in the charge integral of (4.24b) cannot be taken outside the integral sign. Also, we rely on the indirect procedures of Appendices A and B to determine the charge integrals in (4.24) for an arbitrarily moving shell, rather than transform the proper-frame binding force per unit charge (4.22) to obtain the general binding force per unit charge fb (r, t) in an arbitrary inertial frame. The reason for this indirect procedure is that (4.22) holds for diﬀerent spatial points within the shell at one instant of time only in the proper frame, but the relativistic transformation of (4.22) to an arbitrary inertial frame for diﬀerent spatial points within the shell requires the force over an interval of time in the original (proper) frame of reference, even as the radius a approaches zero, because of the 1/a2 term in (4.22). Equations (4.24a) and (4.24b) critically conﬁrm that the rigorously derived binding forces for charge on a relativistically rigid insulator moving with arbitrary center velocity, like the original Poincar´e binding forces (4.2), remove the discrepancy (2.6) between the power equation of motion (2.4) and the force equation of motion (2.1), while having no eﬀect (except for the addition of the mass of the uncharged insulator) on the force equation of motion (2.1), or the 4/3 factor in the electromagnetic mass. With the addition of the binding stresses to the self electromagnetic stresses, the force-power, but not the momentum-energy, transforms as a four-vector; see Section 6.1.

32

4 Internal Binding Forces

4.2.1 Electric Polarization Producing the Binding Forces One can ﬁnd a particular polarization at the surface of the insulator that will produce the static binding forces derived in the previous section. When the charge is at rest, the electric ﬁeld for the binding forces is given by the ﬁrst term of (4.18) within the shell of charge (a < r < a + δ) and zero everywhere else. An electric polarization that would produce this internal binding electric ﬁeld is given simply by a radial polarization density, P(r), within the thin shell of charge proportional to the binding electric ﬁeld ⎧ ⎨ e(r − a) ˆr , a < r < a + δ 4πδa2 P(r) = (4.25) ⎩ 0 , a+δ a + δ . ⎩ 4π0 r2 In other words, the polarization adds a bound volume charge density (−∇· P) that cancels the free-charge density within the surface layer (a ≤ r < a+δ), and a compensating bound surface charge density (P · ˆr) at the outer surface (r = a + δ). The total charge (free charge plus bound polarization charge) reduces to that of Lorentz’s original free-charge shell model of the electron as the thickness δ of the shell approaches zero. As the thickness δ of the shell approaches zero, the electrostatic energy of formation of the free charge and polarization distribution is the same as the free charge alone; thereby conﬁrming that it is not mandatory for the rest energy and mass contributed by the short-range dipolar binding forces to have values other than zero. For the shell at rest, there is no net force exerted on the free charge by the polarization. (When the charge is moving, the results of Sections 4.1 and 4.2 show that the polarization binding forces supply a net force and power to the free charge given in (4.24).) One can also determine an eﬀective molecular polarizability required to produce the polarization that holds the free charge on the stationary insulator. For a linear, homogeneous, isotropic dielectric insulator, the polarization density is proportional to the local ﬁeld

P P P = αv 0 ET + = αv , a < r < a + δ (4.27) 3 3 where the proportionality constant αv is the molecular polarizability per unit volume [13, ch. 2]. This last equation shows that the eﬀective molecular polarizability at the surface of the insulator must be equal to 3.0 in order for

4.2 Binding Forces at Arbitrary Velocity

33

the free-charge distribution to excite the required polarization density. This value of eﬀective molecular polarizability corresponds to an inﬁnite electric susceptibility at the surface of the insulator. In brief, the free charge uniformly distributed in a thin layer at the surface of the insulator induces an eﬀective dielectric polarizability of 3.0 within this layer and a polarized ﬁeld that cancels the self repulsive forces that the free charge exerts on itself. Opposite forces are, of course, exerted on the polarized material of the dielectric insulator. The insulator material does not ﬂy apart because it is held together in the stable energy conﬁgurations described by nonclassical physics (quantum mechanics rather than the classical electrodynamics employed here). Inside the sphere (r < a) it is assumed throughout the book that the values of the permittivity and permeability are equal to those of free space, so that the speed of electromagnetic wave propagation inside as well as outside the sphere is equal to c. Before leaving this chapter on the internal binding forces, let us summarize with hindsight the origin and elimination of the discrepancy in power (2.6) between the Lorentz-Abraham force and power equations of motion. When the charged sphere is stationary, each element of the charge experiences a repulsive force due to its own electric ﬁeld. This electrostatic force integrated over the charge contributes nothing to the total force on the charge. When the charged sphere moves, this static self force transforms relativistically, but still integrates to give a zero total force. However, the moving charged sphere contracts relativistically in the direction of the velocity by an amount proportional to 1/γ, while the component of the static self force per unit charge parallel to the velocity remains unchanged. Thus, the component of the static self force parallel to the velocity does work at a rate proportional to the time rate of change of 1/γ, as exhibited by the negative of the power in (2.6). In addition to the self electrostatic force on the stationary charge distribution, each element of charge is held to the insulator by a binding force that just cancels the electrostatic force. When the charged sphere moves, this binding force exerted on the charge contributes exactly the negative of the power delivered to the charge by the electrostatic force, thereby canceling the discrepancy in power (2.6) between the force and power equations of motion. Lastly, consider a question concerning the mass of the insulator. Even if the rest mass of the insulator is negligible, the insulator exerts a binding force on the charge distribution that does work on the moving charge at the rate given by (2.6) or (4.10). The negative of this binding force is exerted on the insulator by the charge. Consequently, work is done on the moving insulator at the rate given by the negative of (2.6). Thus, one might ask if this work done on the moving insulator changes the total mass of the moving charged sphere. The answer to this question is certainly no because of the Einstein mass-energy relation.. That is, the relativistic mass of the body can be changed by externally applied forces but not by internal forces that do not involve energy that leaves or enters the body. From an alternative viewpoint, the work done by the binding forces exerted on the insulator surface by the

34

4 Internal Binding Forces

charge distribution is canceled by the work done by the stresses inside the insulator material that must oppose the charge-induced binding forces to keep the insulator relativistically rigid.

5 Electromagnetic, Electrostatic, Bare, Measured, and Insulator Masses

As a means of discussing the various masses, let us summarize the basic results that have been derived so far in our re-examination of the Lorentz model of the electron. We begin with a speciﬁc model that we can, in principle, realize in our classical laboratory, namely, a charge e uniformly distributed on the surface of an insulator (insulating material continuum) which remains spherical with constant radius a in its proper inertial frame of reference. Whether or not the model actually approximates the internal structure of the electron is irrelevant to its analysis which is based on Maxwell’s equations with retarded (causal) solutions only, the Lorentz force law, the relativistic generalization of Newton’s second law of motion, the Einstein mass-energy relation, and the short-range polarization forces binding the charge to the insulator surface. When an external force is applied to the shell of charge, for example, by means of an external electric ﬁeld, the charge distribution moves with velocity u(r, t). Except when the insulator has zero velocity, the velocity of the charge at diﬀerent positions r on the surface of the insulator cannot have the same velocity u = u(t) as the center of the insulator if the insulator remains spherical in its proper frame. (The relationship between u(r, t) and the center velocity u(t) is given in equation (A.13) for (u/c)2 1, and equation (B.32) for arbitrary u/c.) The force on each diﬀerential element de of the moving charge is the sum of the externally applied force per unit charge fext (r, t), the internal electromagnetic force per unit charge fem (r, t) generated by the charge itself, and the polarization binding forces per unit charge fb (r, t) holding the charge to the insulator, that is [fext (r, t) + fem (r, t) + fb (r, t)] de .

(5.1a)

Similarly, the work done per unit time by these forces on the element of charge de moving with velocity u(r, t) is [fext (r, t) + fem (r, t) + fb (r, t)] · u(r, t)de .

(5.1b)

36

5 The Masses

The internal self electromagnetic force is determined by the Lorentz force law in terms of the self electric and magnetic ﬁelds excited by the moving charge. The self electromagnetic ﬁelds in the charge distribution derive from Maxwell’s equations with retarded (causal) potentials to give the self electromagnetic force per unit charge in (4.11) in the proper frame. The binding force per unit charge was derived in Section 4.2 by applying Newton’s second law of motion to each diﬀerential element of charge under the requirements that the charge remains uniformly distributed on the relativistically rigid spherical insulator in its proper frame of reference (instantaneous rest frame) and that the mass of the charge distribution is uniformly distributed with the charge in its proper frame. The binding force per unit charge exerted on the charge in the proper frame by the short-range polarization forces holding the charge to the insulator is given in (4.22). It is emphasized that this binding force is not speculated but deduced from the speciﬁc model of the charge residing on the surface of a nonrotating insulator that maintains its spherical shape and uniform charge distribution in its proper frame. The total force F(t) exerted on the charge and the total power P (t) delivered to the charge are found by integrating (5.1a) and (5.1b), respectively, over the charge distribution F(t) = fext (r, t)de + fem (r, t)de + fb (r, t)de (5.2a) charge

charge

P (t) =

charge

fext (r, t) · u(r, t)de +

charge

fem (r, t) · u(r, t)de +

charge

fb (r, t) · u(r, t)de.

charge

(5.2b) By deﬁnition

fext (r, t)de = Fext (t) .

(5.3a)

charge

Also, because the radius a is assumed small enough that the externally applied force varies negligibly with the position over the charge distribution (see (4.13)), the integral involving the external force in (5.2b) becomes fext (r, t) · u(r, t)de = u(t) · fext (r, t)de + O(a2 ) charge

charge

= Fext (t) · u(t) + O(a2 ).

(5.3b)

The expression (B.32) in Appendix B for u(r, t) in terms of the velocity u(t) of the center of the shell has been used to perform the integration in (5.3b). The integrals of the self electromagnetic Lorentz force and power in (5.2a) and (5.2b), shown explicitly in (3.1) and (3.2) and evaluated rigorously in Appendix B for the arbitrarily moving shell of charge, are just the negative of the right-hand sides of the Lorentz-Abraham force equation of motion (2.1)

5 The Masses

37

and the Lorentz-Abraham power equation of motion (2.4), respectively. That is 3γ 2 d e2 e2 γ 2 ¨ + 2 (u · u) ˙ u˙ u fem (r, t)de =− (γu) + 2 3 6π0 ac dt 6π0 c c charge

+

γ2 3γ 2 2 ¨ ˙ (u · u ) + u+ + O(a) (u · u) c2 c2

(5.4a)

and

e2 d fem (r, t) · u(r, t)de =− 6π0 a dt

charge

1 γ− 4γ

(5.4b)

e2 γ 4 3γ 2 2 ¨ ) + 2 (u · u) ˙ + (u · u + O(a) 6π0 c3 c

where, as throughout, u and its derivatives on the right-hand sides of (5.4) refer to the velocity u(t) of the center of the shell. The integrals of the binding force and power in (5.2a) and (5.2b) were evaluated in Section 4.2 and are given explicitly in (4.24a) and (4.24b), respectively. The total binding force (4.24a) involves the mass mins of the insulator, and the power delivered by the binding force (4.24b) to the charge just cancels the electromagnetic power term in the right-hand side of (5.4b) that creates the discrepancy (2.6) between the right-hand side of (5.4b) and u dotted into the right-hand side of (5.4a). Thus, as a result of adding (5.3), (5.4) and (4.24), the total force (5.2a) and power (5.2b) become 3γ 2 d e2 γ 2 ¨ ˙ u˙ F(t) = Fext (t) − (mem + mins ) (γu) + u + (u · u) dt 6π0 c3 c2 γ2 3γ 2 ¨ ) + 2 (u · u) ˙ 2 u + O(a) + 2 (u · u (5.5a) c c and dγ P (t) = F(t) · u(t) = Fext (t) · u(t) − (mem + mins )c2 dt 3γ 2 e2 γ 4 2 ¨ ˙ (u · u ) + + + O(a) (u · u) 6π0 c3 c2

(5.5b)

where the electromagnetic mass mem = e2 /(6π0 ac2 ). Because the binding force has removed the discrepancy between (5.5a) and (5.5b), these two equations can also be written concisely in the four-vector notation given in (2.7). Also, all the information in both (5.5a) and (5.5b) is contained in (5.5a) alone.

38

5 The Masses

5.1 Bare Mass in Terms of Electromagnetic and Electrostatic Masses In (5.5a) we have derived the total force F(t), internal plus external, experienced by the charge moving with arbitrary center velocity u(t). What should this total force equal? One’s ﬁrst thought might be that the total force on the charge should equal the time rate of change of momentum (mes + mins )d(γu)/dt since the electrostatic mass mes is the rest mass of the charge and mins is the rest mass of the insulator; see discussion between (5.8) and (5.9) below. Alternatively, one might think, as Lorentz and Abraham concluded from the measurements of Kaufmann (see Section 5.1.2), that the total force should be zero so that the externally applied force equals the time rate of change of the electromagnetic momentum when mins is zero. But neither of these alternatives is correct if one accepts the relativistic generalization of Newton’s second law of motion [22], [7, sec. 29] that says the total external force applied to a particle should equal, apart from the radiation reaction and forces of order a of a charged particle, the time derivative of momentum of the particle, that is Fext (t) = m

d [γu(t)] − radiation reaction + O(a) dt

(5.6a)

dui radiation i − reaction + O(a) ds

(5.6b)

or in four-vector form i = mc2 Fext

where m is the measured rest mass of the particle (charge plus insulator) and

2 2 3γ 2 radiation = e γ ¨ ˙ u˙ u + (u · u) reaction 6π0 c3 c2 γ2 3γ 2 ¨ ) + 2 (u · u) ˙ 2 u + 2 (u · u (5.6c) c c

2 i

2 j d u radiation i = e i duj du . (5.6d) +u reaction 6π0 ds2 ds ds Accepting the rest mass term in (5.6a)–(5.6b) from the relativistic generalization of Newton’s second law of motion (possible extra terms are discussed in the next subsection 5.1.1), one sees that (5.5) is compatible with (5.6) if the total force in (5.5) equals the time rate of change of momentum F(t) = M0

d (γu) + O(a) dt

(5.7a)

dui + O(a) ds

(5.7b)

or in four-vector form F i = M0 c2

5.1 Bare Mass in Terms of Electromagnetic and Electrostatic Masses

39

with the “bare” mass M0 related to the electromagnetic rest mass (2.2) and the measured rest mass by M0 = m − mem − mins .

(5.8)

Furthermore, the measured rest mass m of the charge shell can be determined theoretically. Assume the charge is initially distributed uniformly on a spherical surface of inﬁnite radius where it has zero mass. The work required to assemble this charge quasi-statically from inﬁnity to the surface of the insulator of radius a is determined from a simple electrostatic calculation [15, sec. 2.7] as e2 /(8π0 a). By the Einstein mass-energy relation, the rest mass of the charged insulator will then be this electrostatic energy of formation divided by c2 , or what is called the electrostatic mass in (2.3), plus the mass mins of the uncharged insulator. (If gravitational ﬁelds [20], [21], or short-range polarization forces binding the charge to the insulator, or short-range forces holding the insulator material together contribute non-negligibly to the rest energy of formation, this mass can be included in mins .) Thus, the measured rest mass of the charged insulator equals the sum of the electrostatic mass and the mass of the insulator m = mes + mins

(5.9)

and the bare mass in (5.8) can be written simply as M0 = mes − mem

(5.10)

or from (2.2) and (2.3) M0 = −

e2 . 24π0 ac2

(5.11)

Emphatically, the value of the bare mass does not depend on the mass mins of the insulator. The ﬁnal form of the equation of motion in (5.6) or, equivalently, in (5.5) and (5.7), can now be written for the charged insulator as 3γ 2 d e2 γ 2 ¨ ˙ u˙ Fext (t) = m (γu) − u + (u · u) dt 6π0 c3 c2 γ2 3γ 2 2 ¨ ) + 2 (u · u) ˙ + 2 (u · u u + O(a) (5.12a) c c 3γ 2 e2 γ 4 2 dγ 2 ¨ ) + 2 (u · u) ˙ Fext · u = mc (u · u + O(a) (5.12b) − dt 6π0 c3 c where m = (mes + mins ) and mes = e2 /(8π0 ac2 ). Of course, (5.12b) is redundant because it is consistent with the equation obtained by taking the dot product of u with (5.12a). The negative bare mass M0 in (5.11) eliminates the 4/3 factor in the inertial rest mass of the original Lorentz-Abraham equation

40

5 The Masses

of motion (2.1) in which the bare mass was assumed zero. With the addition of the bare-mass force-power to the binding and electromagnetic force-power, the total force-power and the total momentum-energy transform as four vectors; see Section 6.1. The mass of the insulator (mins ) allows the equations of motion (5.12) for the charged insulator to be written with an arbitrary value of the inertial rest mass. Thus, (5.12) conforms with the results of Schwinger [23], who shows that stress-momentum-energy tensors with covariant momentum-energy for stable charge-current distributions can be constructed with or without the 4/3 factor. Setting mins equal to mes /3 and zero, respectively, corresponds to Schwinger’s tensors with and without the 4/3 factor; see Section 6.2. (The mass mins can even be negative since, as mentioned above, it includes gravitational and other attractive-force formation energies which, in general, are negative.) The rest mass of the charge has been taken as mes and that of the insulator as mins . In fact, all we can determine experimentally is the total mass m of the charge and insulator together as one body. Ultimately, we do not know where the mass resides—how much in the ﬁeld, in the charge, in the insulator. All we know is the rest mass of the insulator before it is charged and after it is charged. After the insulator is charged, we can ascribe any part of the rest mass to the insulator and the remaining part to the charge, and the above analysis goes through to give the same ﬁnal equations (5.12). Of course, there is a certain appeal and convenience in keeping, as we have above, the rest mass of the charge equal to its electrostatic energy of formation mass mes after the charge is placed on the insulator so that mins refers to the diﬀerence (m − mes ). 5.1.1 Extra Momentum-Energy in Newton’s Second Law of Motion for Charged Particles The relativistic generalization of Newton’s second law of motion in (5.6) for a charged particle is not determined uniquely from the nonrelativistic version of Newton’s second law for uncharged particles. From purely theoretical considerations, any four-vector function of velocity and its time derivatives that vanishes when the charge is zero could be added to the right-hand side of (5.6). If, however, we assume that the only irreversible loss of momentum-energy of the charged particle is the radiated momentum-energy (so that when the initial and ﬁnal velocities and their time derivatives are the same, the only change in momentum-energy will be that which is radiated) then this extra four-vector function must be expressible as the time derivative of a momentum-energy function. In addition, since all the functions in (5.6b) satisfy the condition that their scalar product with ui equals zero (that is, the time rate of change of momentum and energy components of (5.6b) are compatible) the extra function must also satisfy this condition. Thus, on theoretical grounds (5.6b) can be further generalized to [12]

5.1 Bare Mass in Terms of Electromagnetic and Electrostatic Masses i Fext = mc2

dui radiation i dGiextra − reaction + O(a) + ds ds

41

(5.13)

where dGiextra /ds is a four-vector function of velocity and its derivatives that exists only for charged particles and satisﬁes dGiextra ui = 0 . ds

(5.14)

Of course, dGiextra /ds is a function of the charge e, since it vanishes when the charge vanishes and may be a function of the radius a of the charge distribution.1 If we also assume that the only irreversible loss in angular momentum-energy of the charged particle is the radiated angular momentum-energy, since the shell of charge is assumed to translate without rotation, then u × Gextra and its four-vector version, ui Gjextra − uj Giextra , must be expressible as the time derivative of an angular momentum-energy function [24]. This follows from taking the cross product of the position vector r of the center of the particle with the three-vector equation of motion in (5.13) to get r × Fext = m +

d (r × γu) − r × radiation reaction dt

d (r × Gextra ) − u × Gextra + O(a) dt

(5.15a)

or in four-vector form

d j radiation ij i − xj Fext = mc2 (xi uj − xj ui ) − angular xi Fext reaction ds

d xi Gjextra − xj Giextra − ui Gjextra − uj Giextra + O(a). + (5.15b) ds When the initial and ﬁnal position, velocity, and higher derivatives of the position of the center of the particle are the same, the only change in angular momentum will be in the radiated angular momentum if u×Gextra is a perfect time diﬀerential of an angular momentum function (Lextra ) u × Gextra = 1

d Lextra dt

(5.16a)

The reversible and irreversible parts of the radiation reaction term in (5.13) are given in (5.6d) as [e2 /(6π0 )]d2 ui /ds2 and [e2 /(6π0 )]ui (duj /ds) (duj /ds), respectively. In Chapter 8 we show that the series expansions of the self force that lead to this radiation reaction term do not hold during the short proper time interval Δta after the external force is ﬁrst applied. During this transition interval [0, Δta ], the self force and thus the right-hand side of (5.13) may contain a “transition force” (−fai ) with both reversible and irreversible parts that removes the pre-acceleration from the solution to the equation of motion. The O(a) terms in (5.13) also contain both reversible and irreversible parts.

42

5 The Masses

or in four-vector form ui Gjextra − uj Giextra =

d ij L . ds extra

(5.16b)

There is apparently no experimental evidence for the existence of an extra momentum-energy function in the equation of motion of a charged particle at least to order a, and, as Dirac said, “they are all much more complicated than [mc2 dui /ds], so that one would hardly expect them to apply to a simple thing like an electron” [12, p. 154]. Thus, except for the transition force fai explained in Chapter 8 (see Footnote 1 of the present chapter), we will assume Giextra is zero and accept (5.6) as the correct generalization of Newton’s second law of motion for the charged shell. 5.1.2 Reason for Lorentz Setting the Bare Mass Zero All the tools of special relativity [5], [6] were not available to Lorentz and Abraham when they originally derived the total force on the moving Lorentz model of the electron. In particular, the Einstein mass-energy relationship [6] and the relativistic version of Newton’s second law of motion [22] had not appeared. However, Lorentz did assume the pre-relativistic form of Newton’s second law of motion and thus set the total force equal to a constant bare mass M , which Lorentz called the “material” mass, times the acceleration u˙ [4, secs. 28, 32, and 179] to get Fext (t) =

d e2 du radiation (γu) + M − reaction + O(a) . 6π0 ac2 dt dt

(5.17)

(For the charged insulator model, Lorentz’s bare mass M in (5.17) would include the mass of the uncharged insulator, that is, M = M0 + mins .) The key feature of (5.17) is that Lorentz assumed the constant bare mass M in (5.17) was multiplied by du/dt rather than d(γu)/dt (even though he and Abraham had discovered the γ factor in the time rate of change of the electromagnetic momentum in (5.17) before 1905). Between 1901 and 1905, Kaufmann [25] performed experiments to determine the charge to mass ratio for “fast moving” electrons. Lorentz and Abraham hoped that these experiments would decide between Lorentz’s contracting (relativistically rigid) model of the electron and Abraham’s noncontracting (nonrelativistically rigid) model. Although his experiments were not accurate enough to settle this question [26], Kaufmann’s experiments showed clearly that the preponderance of momentum in the electron varied as d(γu)/dt rather than du/dt. Thus Lorentz accepted Kaufmann’s results as experimental evidence that the bare mass in (5.17) was negligible. To quote Lorentz [4, sec. 32], “Of course we are free to believe, if we like, that there is some small material [bare] mass attached to the electron, say equal to one hundredth part of the electromagnetic one, but with a view to simplicity, it will be best to

5.1 Bare Mass in Terms of Electromagnetic and Electrostatic Masses

43

admit Kaufmann’s conclusion, or hypothesis, if we prefer so to call it, that the negative electrons have no material [bare] mass at all. This is certainly one of the most important results of modern physics, . . .” (Abraham also concluded from Kaufmann’s experiments that the bare mass of the electron was zero [3, sec. 16].) As late as 1912, Schott continued to “suppose M zero, in accordance with the most recent measurements” [16, p. 178]. Even after experiments by Bucherer [27] in 1909, Neumann [28] in 1914, and Bohr [29] in 1915 decided in favor of Lorentz’s contracting model over Abraham’s noncontracting model of the electron, and thus also conﬁrmed the prediction of special relativity, at least for “electrical systems,” the bare mass was generally assumed outside the jurisdiction of special relativity and these experiments were regarded as conﬁrming that the bare mass was zero. Richardson [30, ch. 11] summarizes the general consensus in 1915: “These experiments [Bucherer’s] appear to dispose eﬀectually of the rigid [Abraham’s nonrelativistically rigid or noncontracting] electron and they may be regarded as making it reasonably certain that Thomson’s corpuscles are devoid of mass except such as is due to the charge that they carry. For this reason we shall always refer to them in the sequel as negative electrons. We shall ﬁnd later that the relation between [the moving mass] and [the rest mass] characteristic of the Lorentz contractible electron is true of all electrical systems according to the principle of relativity. Bucherer’s experiment may therefore be regarded as evidence in favor of that principle. A remarkable conﬁrmation of the relativity expression for the mass of a moving particle has recently been obtained by N. Bohr from consideration of the decrease of velocity of α and β rays in passing through matter.” Cunningham [31] also gives a very readable account of the conclusions drawn in 1914 from the experiments of Kaufmann et al. By 1920, however, it was generally accepted that the principle of relativity applied to all mass, and Pauli would write, “The old idea that one could distinguish between the constant ‘true’ [bare] mass and the ‘apparent’ electromagnetic mass, by means of deﬂection experiments on cathode rays, can therefore not be maintained” [7, sec. 29]. Thus, one cannot accept (5.17) or continue to assume a bare mass M0 equal to zero, for our speciﬁc model of the electron as a charged insulator, without violating the equivalence of mass and energy and the relativistic version of Newton’s second law of motion, which imply the negative bare mass in (5.11) for this model. Also the bare mass, as pointed out in Section 4.2, should not be confused with the uncharged mass of the insulator. However, because Lorentz’s bare mass corresponds to (M0 + mins ) in our analysis of the charged insulator, Lorentz’s bare mass M can still be zero in the special case when the mass mins of the insulator equals −M0 , that is, mes /3. In that special case the total mass of the charged insulator would be (4/3)mes = mem .

6 Transformation and Redeﬁnition of Force-Power and Momentum-Energy

In chapter 4 it was shown that the speciﬁc model of an electron as a charged insulator demands polarization forces, binding the charge to the insulator, that just cancel the discrepancy (2.6) between the Lorentz-Abraham force and power equations of motion, (2.1) and (2.4). In Chapter 5 we saw that the relativistic generalization of Newton’s second law of motion, together with the Einstein mass-energy equivalence relation, require the negative bare mass in (5.11) that eliminates the factor of 4/3 multiplying the electrostatic mass in the original equation of motion (2.1). In this chapter we summarize the transformation properties of the electromagnetic, binding, and bare-mass force-powers and momentum-energies, derive a total stress-momentum-energy tensor for the charged insulator model of the electron, and review the redeﬁnitions of electromagnetic momentum-energy that have been proposed in the past for the extended electron.

6.1 Transformation of Electromagnetic, Binding, and Bare-Mass Force-Power and Momentum-Energy In order to summarize the transformation properties of the electromagnetic, binding, and bare-mass momentum and energy as well as their time derivatives, force and power, for the charged insulator model of the electron, it will be helpful ﬁrst to make a concise list of these quantities. The self electromagnetic, binding, and bare-mass forces exerted on the charge, and the associated powers delivered to the charge can be written from the preceding chapters as d(γu) dGem 4 = − mes + O(1) dt 3 dt

1 dWem 4 2 d γ− + O(1) =− = − mes c dt 3 dt 4γ

Fem = − Pem

(6.1a) (6.1b)

46

6 Force-Power and Momentum-Energy

d(γu) dGb = −mins + O(a) dt dt

4 1 dγ d dWb = −mins c2 − mes c2 + O(a) Pb = − dt dt 3 dt 4γ Fb = −

(6.2a) (6.2b)

d(γu) dG0 1 (6.3a) = mes dt 3 dt 1 dγ dW0 P0 = − = mes c2 (6.3b) dt 3 dt where the electrostatic mass is given in (2.3), and the dominant parts of the O(1) term in (6.1a) and the O(1) term in (6.1b) are the radiation reaction force and power, respectively. Adding the externally applied force and power to the sum of the electromagnetic, binding, and bare-mass forces and powers in (6.1), (6.2), and (6.3), and setting the total force and power equal to zero give the equations of motion (5.12) for the charged insulator. The momentum and energy of the charged insulator system as a whole can be found by integrating the expressions (6.1), (6.2), and (6.3) of force and power with respect to time for zero initial velocity. For zero initial velocity, the initial electromagnetic momentum, 0 E×BdV over all space, is zero and the binding and bare-mass momenta are zero. The initial electromagnetic energy, (0 /2) (E 2 + c2 B 2 )dV over all space, equals the rest energy of formation of the charge (mes c2 ) and the initial binding energy is chosen equal to the rest energy of the mass of the insulator (mins c2 ). Then, the initial energy of the negative bare mass is zero because the total rest energy of formation of the charged insulator is assumed equal to the sum of the electrostatic and insulator rest energies. (If it is more appealing to have the initial energy of the bare mass equal to −mes c2 /3, one can choose the initial binding energy equal to mins c2 + mes c2 /3. Such a change would add and subtract mes c2 /3 in the following expressions for Wb and W0 , respectively.) The momenta and energies corresponding to (6.1)–(6.3) are F0 = −

4 mes γu + O(1) 3

1 4 u2 + O(1) = mes c2 γ 1 + 2 + O(1) = mes c2 γ − 3 4γ 3c Gem =

Wem

Gb = mins γu + O(a)

1 1 2 2 Wb = mins c γ + mes c − 1 + O(a) 3 γ 1 G0 = − mes γu 3 1 W0 = − mes c2 (γ − 1) . 3

(6.4a) (6.4b) (6.5a) (6.5b) (6.6a) (6.6b)

6.1 Transformation of Force-Power and Momentum-Energy

47

From these expressions of force-power and momentum-energy, one draws the following conclusions about their transformation properties. Neither the electromagnetic momentum-energy (cGem , Wem ) nor its time derivative, the electromagnetic force-power γ(cFem , Pem ), transforms as a four-vector. Similarly, neither the binding momentum-energy (cGb , Wb ) nor the binding forcepower γ(cFb , Pb ) transforms as a four-vector. Also (Fem · u − Pem ) and (Fb · u − Pb ) are not equal to zero. Even the sum of the electromagnetic and binding momentum-energy does not transform as a four-vector. However, the sum of the electromagnetic and binding force-power transforms as a four-vector and satisﬁes (Fem + Fb ) · u − (Pem + Pb ) = 0. The bare-mass forcepower γ(cF0 , P0 ) also transforms as a four-vector satisfying F0 · u − P0 = 0; whereas, the bare-mass momentum-energy (cG0 , W0 ) does not transform as a four-vector, but contributes to the electromagnetic and binding momentumenergy to yield a total momentum-energy that is free of the 4/3 factor and transforms as a four vector. (If, as mentioned above, the initial binding energy were chosen equal to mins c2 + mes c2 /3, so that the initial energy of the bare mass equaled −mes c2 /3, then both the bare-mass momentum-energy and the sum of the electromagnetic and binding momentum-energy would transform as four-vectors.) It may still be disconcerting that the total momentum and energy of a charged massless insulator is not given by the conventional electromagnetic momentum and energy (6.7a) Gem = 0 E × B dV all space

Wem =

0 2

(E 2 + c2 B 2 )dV

(6.7b)

all space

or that the total momentum of a charged massless insulator is not given by the conventional electromagnetic momentum alone even when the velocity of the charge is much less than the speed of light, but contains also the momentum of a negative bare mass. However, one can take consolation in realizing that no law of physics is violated by the conventional electromagnetic momentum not equaling the total momentum of the charge. What we know from Einstein’s mass-energy relation and the relativistic version of Newton’s second law of motion is that the total momentum equals (in addition to the radiation momentum) the electrostatic mass (mes , rest energy of formation divided by c2 ) times the velocity (γu). However, what we know from Maxwell’s equations and the Lorentz force law is merely that the sum of the external and self electromagnetic forces on the charge is Fext − (d/dt)0 E × BdV . Only if this force on the charge equals zero, can the total momentum of the particle be given entirely by the conventional electromagnetic momentum. Since (d/dt)0 E × BdV equals (4/3)mes d(γu)/dt (plus radiation terms) rather than mes d(γu)/dt, the Einstein mass-energy relation and Newton’s

48

6 Force-Power and Momentum-Energy

second law for relativistic motion demand that this force not be zero but equal (−1/3)mes d(γu)/dt, and consequently, that the total momentum of the moving charge not be equal to its conventional electromagnetic momentum alone. From the standpoint of the electromagnetic stress-momentum-energy tensor, it is not surprising that the conventional electromagnetic momentum-energy does not represent the total momentum-energy of the moving charge distribution. Because the electromagnetic stress-momentum-energy tensor is not divergenceless when charge-current is present, the associated momentumenergy will not, in general, be a four-vector. Thus the electromagnetic momentum-energy could not, in general, be expected to represent the total momentum-energy of the system. 6.1.1 Total Stress-Momentum-Energy Tensor for the Charged Insulator The four-divergence of the electromagnetic stress-momentum-energy tensor ij Tem (r, t) equals the force-power density [13], that is

where

ij ∂Tem i = −fem ∂xj

(6.8)

i ≡ ρ(r, t) [fem (r, t), fem (r, t) · u(r, t)/c] fem

(6.9)

ij can be written out as and Tem ij Tem

¯ em ≡ 0 T

¯ em cgem −T = cgem wem

EE − ¯IE 2 /2 + c2 BB − ¯IB 2 /2

(6.10) (6.11a)

(6.11b) gem = 0 E × B 0 wem = (E 2 + c2 B 2 ) . (6.11c) 2 One can also construct stress-momentum-energy tensors with divergences equal to the binding and bare-mass force-power densities, that is ∂Tbij = −fbi (r, t), fbi ≡ ρ(r, t) [fb (r, t), fb (r, t) · u(r, t)/c] ∂xj ∂T0ij e d(γu) dγ i i . = −f (r, t), f ≡ ρ(r, t) , c 0 0 ∂xj 24π0 ac2 dt dt

(6.12) (6.13)

(As usual, when u = u(t) appears without the functional dependence (r, t), it refers to the velocity of the center of the charged shell.) Adding the binding and bare-mass tensors to the electromagnetic tensor would then produce a

6.1 Transformation of Force-Power and Momentum-Energy

49

total stress-momentum-energy tensor whose momentum-energy density would form a four-vector when integrated over all space. Taking the time rate of change of this four-vector momentum-energy produces a four-vector forcepower that, when set equal to the externally applied force, results in the force and power equations of motion (5.12). If no external force is applied to the charged insulator, so that its velocity is constant, the total stress-momentumenergy tensor is divergenceless and the associated four-vector momentum-energy is conserved. First, let us construct the bare-mass tensor T0ij , from its following threevector equations corresponding to (6.13) d(γu) e ¯ 0 + ∂g0 = − −∇ · T ρ ∂t 24π0 ac2 dt c∇ · g0 +

e dγ 1 ∂w0 =− ρ . c ∂t 24π0 ac dt

(6.14a) (6.14b)

A fairly obvious solution to (6.14) is g0 = −

e γρu 24π0 ac2

e γρ 24π0 a e ¯0 = γρuu T 24π0 ac2 w0 = −

(6.15a) (6.15b) (6.15c)

or in four-vector notation T0ij = −

e ρui uj . 24π0 aγ

(6.16)

Rohrlich [32, sec. 6-1] includes the bare-mass tensor (6.16) as part of the “cohesion” or binding stress-momentum-energy tensor. However, for the charged insulator model, it seems preferable to separate the bare-mass tensor from the binding tensor, because we found in Chapters 4 and 5 that the binding forces do not make the inertial mass compatible with the rest energy of formation. It is easily shown that the solution (6.15) satisﬁes (6.14), or that (6.16) satisﬁes (6.13); speciﬁcally we have ¯0 = − −∇ · T

e γ[∇ · (ρu)]u 24π0 ac2

∂g0 ∂(γu) e ∂ρ ρ =− + γu ∂t 24π0 ac2 ∂t ∂t e d(γu) =− ρ − γ[∇ · (ρu)]u 24π0 ac2 dt

(6.17a)

(6.17b)

50

6 Force-Power and Momentum-Energy

c∇ · g0 = −

e γ∇ · (ρu) 24π0 ac

1 ∂w0 e ∂γ ∂ρ =− ρ +γ c ∂t 24π0 ac ∂t ∂t e ∂γ =− ρ − γ∇ · (ρu) 24π0 ac ∂t

(6.17c)

(6.17d)

which produce identities when inserted into the left-hand sides of (6.14a) and (6.14b). The binding stress-momentum-energy tensor must satisfy the following three-vector equations corresponding to (6.12) ¯b + −∇ · T c∇ · gb +

e2 ∂gb mins d = ρ (γu) γδ(r0 − a)ˆr0 + 2 4 ∂t 32π 0 a e dt

e2 1 ∂wb mins c dγ = γδ(r0 − a)ˆr0 · u(r, t) + ρ . c ∂t 32π 2 0 a4 c e dt

(6.18a)

(6.18b)

The charge density in the ﬁrst terms on the right-hand sides of (6.18) has been expressed as a function of the static charge density, that is ρ(r, t) = γρ0 (r0 ) = γδ(r0 − a)

e 4πa2

(6.19)

where r0 is given in terms of r at the time t by the Lorentz transformation r0 = (r − rc )⊥ + γ(r − rc )

(6.20a)

and the position rc of the center of the charged shell can be written in terms of the velocity of the center as t rc =

u(t )dt .

(6.20b)

(The subscripts ⊥ and mean perpendicular and parallel, respectively, to the center velocity u(t) at the time t; and δ(x) is the Dirac delta function.) The binding force per unit charge in (6.18) is equal to the exact binding force per unit charge in (4.22) with the ﬁrst term on the right-hand side of (4.22) averaged over the thickness of the shell and generalized to an arbitrary inertial reference frame. The second term on the right-hand side of (4.22), which is present when the velocity of the charge is not constant, is not included in (6.18). Also, the expressions (6.19) and (6.20a) neglect terms of second order and higher in (r − rc ) when the velocity of the charge is not constant; see (B.30). These secondary binding forces are necessary to hold the accelerating charge to the insulator, but they are inconsequential to the integrated force and power because the results of Chapter 4 (speciﬁcally, equations (4.24))

6.1 Transformation of Force-Power and Momentum-Energy

51

show that their integrals over the charge distribution are of O(a). (In principle, Tbij could be modiﬁed to include the secondary binding stresses, but in practice it may be rather tedious to construct the necessary, relativistically invariant modiﬁcation.) A particularly simple solution to (6.18) is mins γρu e

(6.21a)

e2 mins c2 h(a − r0 ) + γρ 2 4 32π 0 a e

(6.21b)

e2 mins h(a − r0 )¯I − γρuu 32π 2 0 a4 e

(6.21c)

gb = wb = ¯b = T

or in four-vector notation Tbij =

e2 mins c2 i j ij h(a − r )g + ρu u 0 32π 2 0 a4 eγ

(6.22)

where g ij is the metric tensor ⎛

g ij

−1 ⎜ 0 ≡⎜ ⎝ 0 0

0 −1 0 0

0 0 −1 0

⎞ 0 0⎟ ⎟ 0⎠ 1

(6.23)

and h is the unit step function. The preceding solution for Tbij can be used to prove immediately that the mins part of the solution in (6.21) satisﬁes (6.18). To see that the remaining ¯ b and ∂wb /∂t for part of the solution in (6.21) satisﬁes (6.18), evaluate ∇ · T that part to get ∂h e2 e2 ∂h ¯b = − ˆ ˆ ∇h(a − r ) = − + r r −∇ · T 0 ⊥ 32π 2 0 a4 32π 2 0 a4 ∂r ∂r⊥ or ¯b = − −∇ · T

∂h e2 ∂h ˆr0 + ˆr0⊥ γ 32π 2 0 a4 ∂r0 ∂r0⊥

e2 ∇0 h · γˆr0 ˆr0 + ˆr0⊥ ˆr0⊥ 32π 2 0 a4 e2 δ(r0 − a)γ r0⊥ ¯ = −ρfb −∇ · Tb = r0 + 32π0 a5 γ =−

or

(6.24a)

and e2 e2 δ(r0 − a) ∂h(a − r0 ) ∂r0 ∂r0 ∂wb e2 ˆr0 · = = − ∇ h · = 0 4 4 4 ∂t 32π0 a ∂t 32π0 a ∂t 32π0 a ∂t

52

6 Force-Power and Momentum-Energy

or since from (6.20)

r0 ∂γ ∂rc ∂r0 ∂γ ∂ =− = −γu + + [γ(r − rc )] = −γu + (r − rc ) ∂t ∂t ⊥ ∂t ∂t γ ∂t having made use of (∂rc /∂t)⊥ = u⊥ = 0, so that (u · r0 ) d 1 ∂r0 = −γu 1 + ∂t u2 dt γ then 1 ∂wb u · r0 d 1 e2 δ(r0 − a)γ ˆ = −ρfb · u (r, t)/c. (6.24b) · u 1 + = r 0 c ∂t 32π0 a4 c u2 dt γ Inserting (6.24a) and (6.24b) into (6.18) shows that the binding stress-momentum-energy tensor in (6.21) indeed satisﬁes its deﬁning equations (6.18), or equivalently, that (6.22) satisﬁes (6.12). Equations (B.32) and (A.21) have been used to prove in (6.24b) that (to order r02 ) u · r0 d 1 = u (r, t) . u 1+ (6.25) u2 dt γ Thus, the time derivative of wb in (6.21b) equals −ρfb · u (r, t) rather than −ρfb · u(r, t). However, the diﬀerence is inconsequential with respect to the integral over all space of the power density, because ρfb · u⊥ (r, t)dV = 0 (6.26) all space

so that

ρfb · u (r, t)dV =

all space

d e2 ρfb · u(r, t)dV = − 24π0 a dt

1 γ

(6.27)

all space

exactly the right value to cancel the discrepancy (2.6) between the electromagnetic force and power. Note that if we had assumed u were constant in our derivation of the binding force tensor, u (r, t) would equal u, and the total power obtained by integrating the power density would erroneously equal zero, that is ρfb · u dV = u · ρfb dV = 0 (6.28) all space

all space

as explained previously in Chapters 3 and 4. (In (6.24) through (6.30) below, the terms involving the mass of the insulator are ignored since they are irrelevant to this discussion.)

6.1 Transformation of Force-Power and Momentum-Energy

53

One can also obtain the result (6.27) by integrating the energy density wb of the binding tensor over all space to get e2 1 (6.29) Wb = wb dV = 24π0 a γ all space

and taking the negative of the time derivative. Note that Wb in (6.29) diﬀers by a constant (e2 /(24π0 a)) from its value in (4.9) or (6.5b) (with mins = 0). This is because W0 calculated from e2 W0 = w0 dV = − γ (6.30) 24π0 a all space

diﬀers from W0 in equation (6.6b) by the negative of the same constant (−e2 /(24π0 a)), so that the sum, Wb + W0 , remains the same whether it is calculated by adding (6.29) and (6.30) or (6.5b) and (6.6b). As mentioned in Section 6.1, an arbitrary constant energy can be added and subtracted from the binding and bare-mass energies, Wb and W0 , respectively, without changing the total energy of formation or the ﬁnal equations of motion of the charged insulator. In summary, a total stress-momentum-energy tensor T ij has been derived for the charged insulator model of the electron. It can be written as the sum of the electromagnetic, binding-force, and bare-mass stress-momentum-energy tensors ij + T ij (r, t) = Tem

e2 c2 ij h(a−r )g + (mins +M0 )δ(r0 −a)ui uj (6.31) 0 32π 2 0 a4 4πa2

with the bare mass M0 equal, of course, to −mes /3 = −e2 /(24π0 ac2 ). In (6.31) the right-hand side of (6.19) has replaced ρ in (6.16) and (6.22), and r0 is given in terms of (r, t) by the general Lorentz transformation (6.20). The four-divergence of T ij produces the time rate of change of the total momentum-energy density, for the charge distribution bound to the insulator, throughout all space and time; speciﬁcally ∂T ij i = −fem − fbi − f0i (6.32) ∂xj

ρ(r, t) e2 d2 ui dui duj duj = (mes + mins )c2 − + O(a) + ui 2 γe ds 6π0 ds ds ds with ρ(r, t) given in (6.19). The integral over all space of −∂T ij /∂xj produces the sum of the electromagnetic, binding, and bare-mass force-powers given previously in equations (6.1) through (6.3) as well as the radiation reaction and higher order electromagnetic force-power terms. In other words, ∂T ij /∂xj integrated over all space yields a four-vector force-power and the consistent equations of motion

54

6 Force-Power and Momentum-Energy

(5.12) for the charged insulator when this integral is set equal to the externally i applied force. Writing the externally applied force fext as i fext = [fext , fext · u/c] =

ij ∂Text ∂xj

(6.33)

we have

∂ ij (T ij + Text ) = 0. (6.34) ∂xj Also, the integral of T i4 over all space produces the four-vector sum of the electromagnetic, binding-force, and bare-mass momentum-energies given in the equations (6.4) through (6.6), plus the four-vector electromagnetic radiationreaction momentum-energy. If the velocity of the charge distribution is constant (no external force applied), the right-hand side of (6.32) is zero, or equivalently, the divergence of T ij is zero, and it thereby yields a conserved four-vector momentum-energy. When the velocity u is a constant the stress-momentum-energy tensor T ij given in (6.31), together with (6.20), is basically the same as Schwinger’s “ﬁrst stress tensor” [23, eq. (42)]. The diﬀerence is due to Schwinger’s tensor having its bare-mass portion distributed throughout the oblate spheroid, whereas we have assumed the bare mass and mass of the insulator are distributed with the thin shell of charge. Of course, the stress tensors of Schwinger are not derived from the detailed analysis of the charged insulator model of the electron, but are constructed by subtracting a charge-current stress tensor, for a charge in uniform motion, from the electromagnetic stress-momentum-energy tensor, so that the divergence of the resulting tensor is zero. The stress tensors of Schwinger are discussed further in the following section.

6.2 Redeﬁnition of Electromagnetic Momentum and Energy A number of authors, beginning apparently with Fermi [33], have suggested that the consideration of speciﬁc binding forces and bare masses could be avoided in obtaining the equation of motion (5.12) by redeﬁning the electromagnetic momentum and energy (and associated stress-momentum-energy tensor) used to determine the self electromagnetic force and power [23], [32], [34]. In particular, they replace the original electromagnetic momentum and energy densities, 0 E × B and 0 (E 2 + c2 B 2 )/2, in the second integrals of (3.1) and (3.2) by new momentum and energy densities, gnew (r, t) and wnew (r, t), such that the total momentum Gnew and energy Wnew (6.35a) Gnew (t) = gnew (r, t)dV all space

6.2 Redeﬁnition of Electromagnetic Momentum and Energy

55

Wnew (t) =

wnew (r, t)dV

(6.35b)

all space

transform as a four-vector, at least when the charge has constant velocity, and satisfy the consistency requirements (5.14) and (5.16b). Moreover, gnew and wnew can be chosen to eliminate the 4/3 factor that arises using the conventional deﬁnition of electromagnetic momentum and energy. For example, if the stress-momentum-energy tensor is redeﬁned so that the momentum density gnew (r, t) equals γ 2 u multiplied by any invariant function involving the electromagnetic ﬁeld, charge-current, or both [15, sec. 1.23] (invariant with respect to all inertial frames moving with constant relative velocities), and the energy density wnew (r, t) equals γ 2 c2 times the same invariant, that is gnew (r, t) = γ 2 uI (6.36a) wnew (r, t) = γ 2 c2 I

(6.36b)

where u is the velocity of the charge, and I is the invariant, then the total momentum and energy in (6.35) of a charge distribution moving with constant velocity transform as a four-vector. The total momentum and energy in (6.35) calculated from (6.36) determine a four-vector because (γu, γc) is a four-vector and IγdV over all space is an invariant, provided I is calculated for a charge distribution moving with constant velocity. Rohrlich et al. [32], [34] redeﬁne the momentum-energy to yield the speciﬁc invariant 0 I = 2 E 2 − c2 B 2 (6.37) 2c which can be inserted into (6.36) and integrated in (6.35) for a uniformly charged sphere moving with constant velocity u to give the four-vector Gnew (t) = mes γu

mes =

0 2c2

(6.38a)

wnew (t) = mes γc2 (E 2 − c2 B 2 )γdV =

(6.38b) 2

e . 8π0 ac2

(6.38c)

all space

For a charged sphere moving with arbitrary velocity u, (6.37) still yields (6.38) for the dominant 1/a terms of the momentum and energy. Thus when one replaces 0 E × B and 0 (E 2 + c2 B 2 )/2 in the self electromagnetic force and power equations (3.1) and (3.2) by gnew and wnew in (6.36a) and (6.36b), with I inserted from (6.37), the 1/a terms in the ﬁnal forms of the force and power equations of motion, (5.12a) and (5.12b), emerge without the explicit introduction of binding forces or a nonzero bare mass. However, for arbitrary velocity u the invariant (6.37) does not predict the correct radiation reaction terms in the equations of motion (5.12).

56

6 Force-Power and Momentum-Energy

Alternative momentum and energy densities to (6.36) can be found that produce consistent results for the 1/a terms (consistent with the requirements (5.14) and (5.16b) on the rate of change of linear and angular momentumenergy) and correct radiation reaction terms in the momentum and energy equations of motion. Probably the simplest way to do this is to subtract the momentum-energy density (gs , ws ) from the original electromagnetic momentum-energy density 0 [E × B, (E 2 + c2 B 2 )/2] to form gnew = 0 E × B − gs

(6.39a)

0 2 (E + c2 B 2 ) − ws 2 = 0 E × BdV − gs dV

wnew = such that Gnew

all space

and Wnew

0 = 2

(6.39b)

(6.40a)

all space

(E + c B )dV − 2

2

2

all space

ws dV

(6.40b)

all space 2

will form the four-vector (ms γu, ms γc ), that is Gnew = ms γu

(6.41a)

Wnew = ms γc2

(6.41b)

when the charge has constant velocity, where ms is an arbitrary constant mass. For a relativistically rigid charged sphere moving with constant velocity, we see from Appendix B or (6.4a)–(6.4b) that 4 (6.42a) 0 E × BdV = mes γu 3 all space

and

0 2

(E 2 + c2 B 2 )dV =

4 1 mes c2 (γ − ) 3 4γ

(6.42b)

all space

which combine with (6.40) and (6.41) to show that gs and ws must satisfy

4 mes − ms γu (6.43a) gs dV = 3 all space

ws dV =

4 mes c2 mes − ms γc2 − . 3 3γ

(6.43b)

all space

Moreover, if (gs , ws ) are chosen to satisfy (6.43a) and (6.43b) for arbitrary velocity u, then the time derivative of (6.40a) and (6.40b) for arbitrary velocity

6.2 Redeﬁnition of Electromagnetic Momentum and Energy

57

will yield 1/a terms consistent with (5.14) and (5.16b), and correct radiation reaction terms (and all higher electromagnetic terms) in the self force and power. Schwinger [23] has derived divergenceless stress-momentum-energy tensors for constant velocity charge-current distributions, such that a (gs , ws ) can satisfy (6.43) for ms = mem , or (gs , ws ) can satisfy (6.43) for ms equal to the electrostatic mass mes . And, in fact, his method can be immediately generalized to ﬁnd a (gs , ws ) that will satisfy (6.43) for an arbitrary value of the constant mass ms in the 1/a term of the redeﬁned momentum-energy given by (6.40). Speciﬁcally, Schwinger rewrites the electromagnetic force-power density for uniformly moving (constant velocity) charge distributions, that are spherically symmetric in their rest frames, as the divergence of a tensor that depends only on the charge-current distribution. When this force-power tensor, which is not unique, is subtracted from the electromagnetic stress-momentum-energy tensor, a new divergenceless stress-momentum-energy tensor results for which the total momentum-energy is a four-vector. In particular, he ﬁnds the two stress-momentum-energy tensors

and

ij T1ij = Tem + (g ij − ui uj )T

(6.44a)

ij T2ij = Tem + g ij T

(6.44b)

where T is a scalar that depends on the spherical charge distribution. (The ﬁrst is found by subtracting the tensor ui uj T , which is divergenceless at constant velocity, from the second.) For the uniformly moving shell of charge T =

e2 h(a − r0 ) 32π 2 0 a4

(6.44c)

with r0 given in terms of (r, t) through the Lorentz transformation. Thus, the ﬁrst tensor (6.44a) is essentially the same as the stress-momentum-energy tensor (6.31) derived for the charged insulator model when the mass of the insulator mins is zero. Its mass, determined by the integral of the energy or momentum over all space, equals the electrostatic mass. (As mentioned in Section 6.1, the slight diﬀerence between (6.44a) and (6.31) with mins zero is the result of the bare-mass portion of Schwinger’s tensor being distributed throughout the oblate spheroid rather than in the thin shell of charge.) The mass associated with the second tensor (6.44b) equals the electromagnetic mass. It would correspond to a charged insulator with the mass of the insulator material equal to 1/3 the electrostatic mass. Basically, the method of Schwinger is a mathematical way to include the eﬀect of Poincar´e stresses without relating them to the physics of the charged particle. Of course, there are drawbacks to redeﬁning the electromagnetic momentum and energy. If the momentum and energy densities are changed in the

58

6 Force-Power and Momentum-Energy

second integrals of (3.1) and (3.2), so as to also change the values of the time derivatives of these integrals, these new values of self electromagnetic force and power will no longer equal the Lorentz force and power (the ﬁrst integrals in (3.1) and (3.2)) for the shell of charge. This implies one or more of the following alternatives: 1. 2. 3. 4.

the deﬁnition of the Lorentz force must change Maxwell’s equations must change the charge-current distribution must change unknown forces (electromagnetic or nonelectromagnetic) are present that contribute to the total self force and power of the charge distribution.

None of these alternatives seem very attractive because they each involve introducing extra unknowns unnecessarily into the simple, deterministic model of the electron as an insulator that remains spherical and uniformly charged in every proper inertial frame of reference. Also, none of the redeﬁned stressmomentum-energy tensors predict the second and higher order binding forces on the right-hand side of (4.22) that are necessary to hold the accelerating charge to the insulator.

7 Momentum and Energy Relations

The equations of motion (5.12) for the charged insulating sphere of radius a moving with arbitrary center velocity u(t) can be rewritten in four-vector notation [13] as

2 i j e2 d u dui i i duj du − + O(a) (7.1) Fext = mc2 + u ds 6π0 ds2 ds ds with

i Fext ≡ γ (Fext , Fext · u/c) i

(7.2a)

u ≡ γ (u/c, 1)

(7.2b)

ui ≡ γ (−u/c, 1)

(7.2c)

ds ≡ cdt/γ .

(7.2d)

The measured rest mass m of the charged insulator equals (mes + mins ) with mes = e2 /(8π0 ac2 ). The total momentum G12 and energy W12 supplied by the external force to the charge between the times t1 and t2 are given by t2 G12 =

Fext (t)dt

(7.3a)

Fext (t) · u(t)dt

(7.3b)

t1

and

t2 W12 = t1

or in four-vector notation Gi12

s2 = (cG12 , W12 ) = s1

i Fext ds .

(7.4)

60

7 Momentum and Energy Relations i Substituting Fext from (7.1) into (7.4) we obtain i du e2 dui (s2 ) − (s1 ) Gi12 = mc2 ui (s2 ) − ui (s1 ) − 6π0 ds ds s2 duj duj e2 ds + O(a) . ui − 6π0 ds ds

(7.5)

s1

If the velocity and acceleration of the particle are the same at times t1 and t2 , that is, at s1 and s2 , the momentum-energy in (7.5) reduces to Gi12

e2 =− 6π0

s2 s1

ui

duj duj ds + O(a) . ds ds

(7.6)

In three-vector notation ui

4 γ duj duj γ6 ˙ 2 + 7 (u · u) ˙ 2 (u, c) = −γ 5 |u| ds ds c c

(7.7)

so that (7.6) becomes Gi12 = (cG12 , W12 ) t2 γ6 e2 4 2 2 ˙ ˙ γ (u, c)dt + O(a) . | u| + (u · u) = 6π0 c4 t1 c2

(7.8)

The integrand in (7.8) is just the momentum and energy radiated per unit time by an accelerating point charge [35], [3, sec. 15]. Thus, (7.8) says that the momentum and energy imparted to the charge by the externally applied force during any time interval is equal to the momentum and energy radiated by that charge, provided the initial and ﬁnal velocities and accelerations are the same. In other words, the dui /ds and d2 ui /ds2 terms in the equation of motion (7.1) represent reversible rates of change of momentum-energy, while the ui duj /ds duj /ds term represents the irreversible rate of change of momentum-energy that radiates to the far ﬁeld. The reversible dui /ds term is, of course, the usual rate of change of momentum-energy four-vector in the relativistic version of Newton’s second law of motion dui d mc2 = mγ (γu, γc) . (7.9) ds dt Its integral over a proper time interval determines the reversible change in kinetic momentum-energy of the particle during that time interval. The reversible d2 ui /ds2 term can be written in three-vector form as e2 γ d 1 4 e2 d2 ui γ4 2 ˙ ˙ ˙ γ = − (u · u)u , u · u) . (7.10) − u + (γ 6π0 ds2 6π0 c3 dt c2 c

7 Momentum and Energy Relations

61

When this perfect diﬀerential is integrated over proper time it yields a reversible change in momentum-energy that cannot be classiﬁed as either a change in kinetic momentum-energy or a change in radiated momentum-energy (which is irreversibly lost to the far ﬁeld). Schott [36] called the energy portion of (7.10), that is e2 γ 4 ˙ − (u · u) (7.11) 6π0 c4 the “acceleration energy” because it “must be regarded as work stored in the electron in virtue of its acceleration.” Therefore, this part of (7.10) is sometimes referred to as the Schott energy term, although Abraham [3, sec. 15] had previously separated the reversible momentum as well as the reversible energy of (7.10) in his derivation of the radiation reaction for a charge moving with arbitrary velocity. Before and after the external force is applied, the acceleration of the charge is zero (ignoring the pre-acceleration that will be discussed in Chapter 8) so that the Schott acceleration momentum-energy is zero and, as expected, the momentum-energy that has been supplied by the external force has been converted entirely to kinetic and radiated momentum-energy. However, while the external force is being applied, the charge is accelerating and the momentum-energy supplied by the external force is converted to “Schott acceleration momentum-energy,” as well as kinetic and radiated momentum-energy. A physically intuitive understanding of the “acceleration” momentum-energy can be gained by looking at (7.1) for time harmonic motion. With the help of (7.7), (7.9) and (7.10), the momentum and energy equations of motion in (7.1) may be written separately in three-vector notation as e2 d d(γu) γ4 2 ˙ ˙ Fext = m − γ (u · u)u u + dt 6π0 c3 dt c2 γ4 γ2 2 2 ˙ ˙ − 2 |u| + 2 (u · u) u + O(a) (7.12a) c c and

d 4 γ2 ˙ 2 + 2 (u · u) ˙ − γ 4 |u| ˙ 2 + O(a). (γ u · u) dt c (7.12b) The ﬁrst terms on the right-hand sides of (7.12) can be interpreted simply as the rates of change of kinetic momentum and energy required to accelerate the energy of formation of the moving charge. To understand the second terms on the right-hand sides of (7.12), consider a charge oscillating rectilinearly with sinusoidal frequency ω, so that the velocity is given by Fext · u = mc2

dγ e2 − dt 6π0 c3

u(t) = U0 sin(ωt)

(7.13)

and the radiation reaction terms in the energy equation of motion (7.12b) become

62

7 Momentum and Energy Relations

d ˙ = −U02 ω 2 cos(2ωt) − (γ 4 u · u) dt γ2 ˙ 2 + 2 (u · u) ˙ 2 = U02 ω 2 cos2 (ωt) γ 4 |u| c

(7.14) (7.15)

when (u/c)2 1. The irreversible reaction term (7.15) behaves as a time-harmonic radiated power, that is, it has the time dependence of the Poynting vector integrated over a closed surface in the far ﬁeld. Its average over time has the positive value U02 ω 2 /2. The reversible “acceleration” reaction term (7.14) behaves as a reactive power whose average over time is zero. In other words, if the oscillating charge were an antenna fed by a single-frequency input voltage and current, (7.14) and (7.15) would contribute to the reactive and resistive (radiation resistance) parts, respectively, of the input impedance of the antenna [37]. (The time rate of change of the kinetic energy of the oscillating charge has a time dependence proportional to U02 ω sin(2ωt) ,

(u/c)2 1

(7.16)

and thus it would also contribute to the reactive part of the input impedance of the antenna.) For a charge whose velocity and acceleration are continually increasing with time, rather than oscillating, the reversible kinetic energy continually increases, the irreversible radiated power increases, and more and more reversible Schott acceleration energy is taken from the electromagnetic ﬁelds of the charge. A similar unlimited increase in the irreversible (radiated) and reversible (reactive kinetic and Schott) energies occurs when the frequency of an oscillating charge or electric dipole is continually increased, as one can see from (7.14)–(7.16). However, the reactive energy taken from the oscillating charge, although it can increase without limit by increasing the frequency, is always returned to zero and supplied to the charge in an equal amount during each half period of oscillation. 2 ˙ + γ 2 (u · u) ˙ 2 /c2 Before leaving this section, note that the quantity γ 4 |u| can be substituted from (7.12b) to simplify the form of the equation of motion (7.12a) to γ 2 [Fext − (Fext · u)u/c2 ] = mγ 3 u˙ −

e2 γ d 3 ˙ + O(a). (γ u) 6π0 c3 dt

(7.17a)

This form of the force equation of motion comes in handy in the next section of this chapter and in Chapter 8. The power equation of motion (7.12b) can be recast into the form 2

e2 γ d γ γ3 2 2 dγ ˙ − Fext · u = mc + O(a). (7.17b) − 2 |u| dt 6π0 c dt2 c

7.1 Hyperbolic Motion

63

7.1 Hyperbolic Motion For relativistically uniform acceleration, which is deﬁned as [32, secs. 5-3 and 6-11] duj duj d2 ui + ui =0 (7.18) 2 ds ds ds the reversible Schott reactive power cancels the radiated power and the equation of motion (7.1) reduces to that of an uncharged particle i Fext = mc2

dui + O(a) ds

(7.19)

that is, the time rate of change of kinetic momentum-energy equals the applied force minus the O(a) terms. Because in the proper inertial reference frame ¨ = 0, that is, u˙ = constant, it is appropriate to (7.18) reduces to simply u refer to the motion deﬁned by (7.18) as relativistically uniform acceleration. The charged particle radiates by drawing energy from the reactive ﬁelds of the charge, the reactive ﬁelds continually being replenished by the increasing acceleration of the charge. Neglecting terms of O(a), equation (7.19) reduces to the Newtonian equation of motion in four-vector form i Fext = mc2

dui . ds

(7.20)

Relativistically uniform acceleration or “hyperbolic motion” is described in numerous physics texts [32]. Here we show that the equations (7.18) and (7.20) cannot both be satisﬁed unless there is an inertial frame in which the motion is rectilinear and the externally applied force is constant, such as when a charge is placed in a uniform electrostatic ﬁeld. To prove this result, use the procedure for deriving (7.17a) to express the three vector part of (7.18) as simply d 3 ˙ =0 (γ u) dt

(7.21)

and write the three-vector part of (7.20) as Fext = m

d (γu). dt

(7.22)

The solution to (7.21) is γ 3 u˙ = A0

(7.23)

where A0 is a constant vector. The result in (7.23) implies that the acceleration of the charge perpendicular to the direction of A0 is zero. Consequently, the velocity of the charge is constant in the direction perpendicular to the direction of A0 . In the inertial reference frame moving with this constant velocity, the motion is rectilinear. Moreover, (7.18) and thus (7.21) still hold in

64

7 Momentum and Energy Relations

this inertial reference frame in which the motion is rectilinear because (7.18) is a Lorentz invariant expression. Therefore, in this rectilinear reference frame, (7.22) and (7.23) become d Fext = m (γu) (7.24) dt and d γ 3 u˙ = (γu) = A0 . (7.25) dt A comparison of (7.24) and (7.25) immediately reveals that Fext = mA0 .

(7.26)

In other words, relativistically uniform motion can only be achieved if there exists an inertial reference frame in which the motion is rectilinear and the externally applied force is equal to a constant, for example, an electron accelerated by an electrostatic ﬁeld E0 between two oppositely charged parallel plates that are normal to the direction of motion of the charge. Except during the short transition intervals when the electron enters and leaves the parallel plates, the motion of the electron is determined entirely by the relativistic Newtonian equation of motion eE0 = m

d (γu). dt

(7.27)

The problem of the motion of a charge between two parallel plates, including its behavior during the entrance and exit transition intervals, is considered in Section 8.2.4 after the equation of motion is modiﬁed to eliminate preacceleration and pre-deceleration.

7.2 Runaway Motion “Runaway solutions” are homogeneous solutions to (7.1). In other words, the Schott reactive power cancels both the radiated power and the kinetic power so that (7.1) is satisﬁed without an applied external force j d2 ui 6π0 mc2 dui i duj du = −u + ds2 ds ds e2 ds

(7.28)

(neglecting O(a) terms), or in three-vector notation 6π0 mc3 d(γu) γ4 γ2 γ4 d 2 2 2 ˙ + 2 (u · u) ˙ ˙ γ u˙ + 2 (u · u)u u+ = 2 |u| (7.29a) dt c c c e2 dt 6π0 mc5 dγ γ2 d 4 4 2 2 ˙ + 2 (u · u) ˙ = γ |u| ˙ . (7.29b) + (γ u · u) dt c e2 dt

7.2 Runaway Motion

65

A formal, general solution to the homogeneous equation of motion (7.29a) can be found by rewriting this equation in the shortened form given in (7.17a), namely d 3 e2 1 ˙ − γ 2 u˙ = 0 , (7.30) τe = (γ u) dt τe 6π0 mc3 Recasting this equation as d 3 1 ˙ − ˙ =0 (γ u) (γ 3 u) dt τe γ

(7.31)

allows us to solve for γ 3 u˙ as ⎛ 1 γ 3 u˙ = A1 exp ⎝ τe

t

⎞ dt ⎠ + A2 γ(t )

(7.32)

0

wherein A1 and A2 are arbitrary constants. Since 1/γ(t) = 1 − u2 (t)/c2 , the integral in (7.32) approaches a positive inﬁnite value unless u(t) → c. Therefore, as t → ∞, the speed of the particle “runs away” toward the speed of light, even though there is no external force applied. Schott derived the runaway motion as part of his solution to the equation of rectilinear motion in his 1915 paper [36, eq. (21)]; see Section 8.1. He called this motion “exponential motion” and argued “that the exponential motion is not realizable experimentally, at any rate not with electric ﬁelds at our command.” Although these runaway solutions are presumably not physically realizable, they are mathematically valid homogeneous solutions to the diﬀerential equation of motion that do not violate conservation of momentum-energy. Both the increasing reversible kinetic momentum-energy and the increasing irreversible radiated momentum-energy are taken entirely from the reservoir of reversible Schott reactive momentum-energy that is continually being supplied by the increasing acceleration of the charge. It is emphasized that the unlimited supply of Schott reactive momentum-energy for the runaway modes is produced by the unlimited increase in the four-acceleration of the particle, and is not dependent upon the radius of the charge approaching zero or mass of the particle approaching inﬁnity. Despite the fact that the homogeneous runaway solutions do not violate the conservation of momentum-energy, it is shown in Chapter 8 that the runaway behavior is eliminated from the complete solution of (7.1) by invoking the asymptotic condition of zero acceleration as t approaches inﬁnity. In an attempt to get an equation of motion that involves only the kinetic and radiated momentum-energy of a charged particle, one may be tempted to simply discard the reactive momentum-energy term in (7.1) or its threevector equivalent in (7.12). Unfortunately, the resulting simpliﬁed equation i of motion would no longer be consistent with Fext ui equal to zero. In terms

66

7 Momentum and Energy Relations

of the three-vector equations, the scalar product of u with (7.12a) would no longer equal (7.12b). It seems quite remarkable that without the insight and transformations of special relativity, Abraham was able to determine the reversible (reactive) parts of the radiation reaction force and power in (7.12) from a knowledge of the radiated momentum and energy of an accelerating point charge; then prove that the solution was unique [3, sec. 15]. (In the four-vector notation of (7.1) and with the transformations of special relativity, the determination of the reversible part of the radiation reaction from the radiated part is an elementary exercise. Uniqueness of solution follows from the fact that a fourvector which reduces to zero in the proper inertial frame must be zero in an arbitrary inertial frame.)

8 Solutions to the Equation of Motion

As a preliminary to solving the equation of motion (7.1) for the uniformly charged sphere of radius a and total charge e, write the magnitude of the four acceleration in (7.1) as (w · w )2 duj duj w2 = − 4 2 6 ds ds γ c c

(8.1a)

where w is deﬁned in terms of the velocity of the center of the shell by w = γu,

γ = (1 − u2 /c2 )−1/2 = (1 + w2 /c2 )1/2

(8.1b)

and the primes denote derivatives with respect to the proper time dτ = dt/γ . Insertion of (8.1) into (7.1) yields the three-vector equation for w

e2 1 (w · w )2 2 γFext (τ ) = mw − w − 2 w − w + O(a). 6π0 c3 c c2 γ 2

(8.1c)

(8.2)

For rectilinear motion in the x direction Fext (τ ) = Fext (τ )ˆ x

(8.3a)

w = wˆ x

(8.3b)

and (8.2) becomes γFext (τ ) = mw −

w2 w e2 1 + O(a) . w − 6π0 c3 c2 (1 + w2 /c2 )

(8.4)

(In (8.2)–(8.4) and throughout the rest of this chapter, the vector and scalar functions Fext (τ ) and Fext (τ ) are compact notations for Fext [t(τ )] and Fext [t(τ )], respectively.) Following Schott [36] we see that the substitution

68

8 Solutions to the Equation of Motion

w/c = sinh(V/c)

(8.5)

reduces this equation for rectilinear motion to the simpler diﬀerential equation Fext (τ ) O(a) = V (τ ) − τe V (τ ) + m m where τe =

(8.6)

e2 . 6π0 mc3

(8.7)

If m and e equal the mass and charge of an electron, the time interval τe is equal to the amount of time it takes light to travel four thirds the classical radius of the electron.

8.1 Solution to the Equation of Rectilinear Motion If the O(a)/m terms in (8.6) are neglected, the most general solution to the resulting equation of rectilinear motion can be written as ⎡ ⎤ τ 1 V (τ ) = − exp(τ /τe ) ⎣ Fext (τ ) exp(−τ /τe )dτ + A⎦ , mτe 0

−∞ < τ < ∞

(8.8)

where the external force is applied at τ = 0 (t = 0) and is assumed zero for all time τ < 0 (t < 0). Integration of (8.8) with respect to the proper time τ gives the general solution for V as 1 V(τ ) = B + m

τ 0

Fext (τ )dτ

⎡

1 − exp(τ /τe ) ⎣ m

τ

⎤ Fext (τ ) exp(−τ /τe )dτ + τe A⎦ ,

0

−∞ < τ < ∞ .

(8.9)

Integrating (8.9) with respect to the proper time, one could also obtain the position of the center of the shell. This would introduce a third arbitrary constant (A and B being the other two) that can be determined by specifying the position of the particle at a certain time, or in the remote past. To determine the two remaining constants, A and B, two other boundary conditions are required. This is one more constant and boundary condition than is required by Newton’s second law of motion for uncharged particles, which involves only the ﬁrst derivative of velocity, rather than the ﬁrst and second derivatives in (8.6). At ﬁrst thought, since the external force is not

8.1 Solution to the Equation of Rectilinear Motion

69

applied until τ = 0, one might set the velocity and acceleration equal to zero at τ = 0 to obtain zero for both the constants A and B. Then (8.9) would become 1 V(τ ) = m

τ

exp(τ /τe ) Fext (τ )dτ − m

0

τ

Fext (τ ) exp(−τ /τe )dτ ,

0

−∞ < τ < ∞ .

(8.10)

Unfortunately, there is a serious problem with the solution (8.10). The velocity function V(τ ) and all its derivatives approach inﬁnity [u(t) → c] as τ → ∞, even when the external force is applied for a ﬁnite time. Returning to (8.8) or (8.9) we see that these “runaway solutions” (called “exponential motion” by Schott [36]; see Section 7.2) are eliminated as τ → ∞ if and only if the constant A is given by 1 A=− mτe

∞

Fext (τ ) exp(−τ /τe )dτ .

(8.11)

0

Equation (8.11) ensures that the acceleration in (8.8) approaches zero as τ → ∞, if the external force approaches zero as τ → ∞; and thus (8.11) can be considered a result of the “asymptotic condition” [12], [32] ˙ lim u(t) =0

(8.12a)

lim Fext (t) = 0 .

(8.12b)

t→∞

when t→∞

(Rohrlich [32, sec. 8-2] points out that the asymptotic condition can be based on a fundamental “principle of undetectability of small charge,” which asserts that the motion of a charged particle must approach that of a neutral particle in the limit as the charge approaches zero.) After insertion of A from (8.11), equations (8.8) and (8.9) can be written as 1 V (τ ) = mτe

∞

Fext (τ ) exp[−(τ − τ )/τe ]dτ ,

(8.13a)

τ

−∞ < τ < ∞ and 1 V(τ ) = B + m −∞ < τ < ∞ or

∞ τ

1 Fext (τ ) exp[−(τ − τ )/τe ]dτ + m

τ

Fext (τ )dτ ,

0

(8.13b)

70

8 Solutions to the Equation of Motion

1 V(τ ) = B + m

∞

1 Fext (τ + τ ) exp(−τ /τe )dτ + m

τ

0

Fext (τ )dτ ,

0

−∞ < τ < ∞ .

(8.13c)

A ﬁnal boundary condition is needed to evaluate the constant B in (8.13b)– (8.13c). One can evaluate B by specifying the initial velocity, but this procedure leads to a velocity in the remote past (τ → −∞) that depends on the external force, which we have assumed is applied at τ = 0. Speciﬁcally, if one enforces the initial condition V(0) = 0 in (8.13b)–(8.13c), then both the constant B and the velocity function in the remote past are given by 1 B = V(−∞) = − m

∞

Fext (τ ) exp(−τ /τe )dτ .

(8.14)

0

Physically, it is much more appealing to demand that in the remote past the velocity be zero or a constant that is independent of the applied force. Thus, if the ﬁnal boundary condition on the motion of the charge is an asymptotic condition on the velocity in the remote past; in particular, for zero velocity in the remote past lim u(t) = 0 (8.15) t→−∞

then B = 0 and (8.13c) becomes 1 V(τ ) = m

∞

1 Fext (τ + τ ) exp(−τ /τe )dτ + m

0

1 = m

∞ τ

τ

Fext (τ )dτ

0

1 Fext (τ ) exp[−(τ − τ )/τe ]dτ + m

= τe V (τ ) +

τ

Fext (τ )dτ

0

1 m

τ

Fext (τ )dτ ,

−∞ < τ < ∞ .

(8.16)

0

Equation (8.16), combined with the deﬁnitions (8.5) and (8.1b), is the general solution for the rectilinear velocity u of the center of the shell of charge for all time under the two asymptotic conditions that the acceleration approaches zero in the distant future (for zero external force in the distant future) and the velocity approaches zero in the remote past. Of course, the external force must be well-behaved enough for the integrals in (8.16) to exist, and the solution was obtained under the assumption that the terms of O(a)/m in (8.6) could be neglected. The solution (8.16) exhibits a number of peculiarities. The most unsettling one, pre-acceleration, or acceleration before the external force is applied at τ = 0, is considered in Section 8.2. A second peculiarity, namely that the

8.1 Solution to the Equation of Rectilinear Motion

71

velocity at any instant of time depends on the externally applied force at all future times, is discussed near the end of Section 8.2.2. A third peculiarity with the solution (8.16) is that if the force is zero after it is applied over a ﬁnite time interval, 0 ≤ τ < τ0 , the velocity function reduces to τ0 1 V(τ ) = V(τ0 ) = Fext (τ )dτ , τ > τ0 (8.17a) m 0

the ﬁnal velocity function one would obtain if the radiation term V in (8.6) (as well as the O(a)/m terms) were ignored entirely. This result in (8.17a) is not so objectionable, if one realizes from (8.5) and (8.1b) that ⎤ ⎡ τ0 1 Fext (τ )dτ ⎦ γ(t)u(t) = γ(t0 )u(t0 ) = c sinh ⎣ mc 0

=

1 m

t0

Fext (t )dt ,

t > t0

(8.17b)

0

so it does not imply that the radiated momentum-energy is zero, or that the impulse and work supplied to the charged sphere τ by the external t force is converted to kinetic energy alone. (Note that 0 0 Fext (τ )dτ = 0 0 Fext (t )dt .) To see this, integrate (7.12) over all time that the velocity is changing (−∞ < t < t0 ) to get (for rectilinear motion) t0

t0 Fext dt =

−∞

t0

0

t0

t0 γ 6 u˙ 2 u(t)dt

0

(8.18a)

−∞

e2 Fext udt = mc [γ(t0 ) − 1] + 6π0 c3

t0

2

Fext udt = −∞

e2 Fext dt = mγ(t0 )u(t0 ) + 6π0 c5

γ 6 u˙ 2 (t)dt. (8.18b) −∞

The reversible Schott acceleration momentum-energy in (7.12) does not contribute to (8.18) because the ﬁnal acceleration and the acceleration in the remote past are both zero. The ﬁrst terms on the right-hand sides of (8.18) give the total change in the kinetic momentum-energy of the charged sphere, while the second terms give the total momentum-energy radiated by the charged sphere. During pre-acceleration (−∞ < t < 0) only the runaway solution is present, and, as explained in Chapter 7, the Schott momentum-energy cancels both the kinetic and radiated momentum-energy. If the ﬁnal velocity of the charge also equals zero (u(t0 ) = 0) the change in the kinetic momentum-energy is zero and (8.18) conﬁrms that the entire impulse and work delivered by the external force is converted to radiated momentum-energy. Note that even when the ﬁnal velocity (as well as velocity in the remote past) is zero, we have the inequalities

72

8 Solutions to the Equation of Motion

t0

e2 Fext dt = 6π0 c5

0

and

t0

t0 γ 6 u˙ 2 u(t)dt

(8.19a)

0

e2 Fext udt = 6π0 c3

0

t0 γ 6 u˙ 2 (t)dt .

(8.19b)

0

That is, in order for the total momentum and energy radiated to equal the impulse and work delivered by the externally applied force when the ﬁnal velocity (and velocity in the remote past) are zero, the integration of the radiated time rate of change of momentum and energy must include the preacceleration, because the initial velocity u(0) is not zero in the pre-acceleration solution (8.16). 8.1.1 Formal Solution to the General Equation of Motion A formal solution to the general equation of motion can be found from its form given in (7.17a), which can be rewritten as γ2 d ˙ Fext − (Fext · u)u/c2 = γ 3 u˙ − τe γ (γ 3 u) m dt

(8.20)

with the O(a)/m terms neglected. The solution to this ﬁrst order diﬀerential ˙ under the asymptotic condition in (8.12) is given for all t equation for (γ 3 u), as 1 ˙ γ (t)u(t) = mτe

∞

3

t

γ(t ) Fext (t ) − [Fext (t ) · u(t )] u(t )/c2 ! · exp − (1/τe )

t

" γ −1 (t )dt dt

(8.21a)

t

which can be rewritten as 1 ˙ )= γ (τ )u(τ mτe

∞

3

τ

$ # γ(τ ) Fext (τ ) − [Fext (τ ) · u(τ )] u(τ )/c2 · exp[−(τ − τ )/τe ]dτ

(8.21b)

after making the change of variable dτ = dt/γ. The equations in (8.21) represent a rigorous solution for the acceleration ˙ u(t) in terms of both u(t) and Fext (t). However, because u(t) is unknown, ˙ (8.21) is not a useful solution for calculating u(t) or u(t). Nonetheless, the ˙ solution (8.21) reveals that the acceleration u(t) of the charged sphere does not vanish before the external force is applied (say at t = τ = 0) and thus the

8.2 Cause and Elimination of the Pre-Acceleration

73

solution to the general equation of motion, like the solution to the rectilinear equation of motion derived in Section 8.1, exhibits pre-acceleration. If the external force Fext (t) applied to the charged sphere is zero for t > t0 , (8.21) ˙ also reveals that the acceleration u(t) = 0 for t > t0 , but that the solution anticipates the shutting oﬀ of the external force by displaying pre-deceleration. Both the rectilinear solution in (8.16) and the general solution in (8.21b) also reveal that the pre-acceleration (and pre-deceleration) diminishes as the externally applied force varies more slowly on the scale of τe [38].

8.2 Cause and Elimination of the Pre-Acceleration The solution (8.21) to the general equation of motion and the solution (8.16) to the rectilinear equation of the motion both predict nonzero acceleration before the external force is applied at t = 0 or τ = 0. One may be tempted to simply set the acceleration or velocity equal to zero for τ < 0 to eliminate the pre-acceleration in (8.16), for example. However, the resulting solution does not satisfy the original diﬀerential equation (8.6) (with O(a)/m terms neglected) because delta functions and derivatives of the delta functions are introduced into the derivatives of the velocity at τ = 0. For example, if the external force is a step function applied at τ = 0 0 , τ <0 Fext (τ ) = (8.22) F0 , τ ≥ 0 then the solution (8.16) becomes simply % F0 τe exp(τ /τe ) , τ ≤ 0 V(τ ) = m 1 + τ /τe , τ ≥ 0 .

(8.23)

We see that (8.23) satisﬁes the equation of motion (8.6) (with the O(a)/m terms neglected) for all τ , whereas setting V(τ ) = 0 for τ < 0 in (8.23) violates the equation of motion by introducing delta and doublet functions in V (τ ) and V (τ ) at τ = 0. Similarly, a spurious delta function is introduced into V (τ ) by diﬀerentiating (8.23) and setting the acceleration zero for τ < 0, regardless of the initial velocity.1 (Note that the condition limτ →∞ Fext (τ ) = m limτ →∞ V (τ ) = F0 for this step-function external force in (8.16) replaces the asymptotic condition in (8.12).) Although the noncausal pre-acceleration decays in the past at the rapid rate of 1/(2.718 · · · ) in the proper time interval τe that light takes to travel 4/3 the classical radius of the electron if m and e are the mass and charge of the 1

In Section 8.2.2 we show that a rigorous derivation of the self force near the nonanalytic points in time of the external force (here at τ = 0) allows for deltalike functions that eliminate the noncausal pre-acceleration to be added to the original equation of motion immediately after τ = 0.

74

8 Solutions to the Equation of Motion

electron, it is disconcerting that the pre-acceleration appears in the solution to the equation of motion because the equation of motion is based on causal (retarded-potential) solutions to Maxwell’s equations. It is not surprising that the equation of motion of a charged particle allows homogeneous solutions like the runaway modes, which are not present in Newton’s second law of motion for uncharged particles, because the radiation reaction introduces time derivatives of acceleration into the equation of motion. The disturbing feature of the equation of motion is that when the asymptotic condition in (8.12) is applied to eliminate the runaway modes from the inhomogeneous solution, noncausal pre-acceleration cannot be avoided for a solution that remains wellbehaved at t = τ = 0, the time the external force is ﬁrst applied. The cause of the pre-acceleration solution will be determined by returning to the derivation of the equation of motion of the extended model of the electron. Before doing so, however, let us show that the pre-acceleration is not eliminated by including the higher order terms in the equation of motion (O(a)/m terms in the equation of rectilinear motion (8.6)). The pre-acceleration solution (8.16) is a solution to (8.6) when the O(a)/m terms are negligible. If the mass m is replaced by mes = e2 /(8π0 ac2 ), then one can obtain suﬃcient conditions for the O(a)/mes terms to be negligible in the proper frame (u = 0) by returning to the series expansion for the self electromagnetic force (see (D.17) in Appendix D) and noting that the linear terms of O(a)/mes in the proper-frame version of (8.6) are negligible if the sum

n ∞ 1 dn u(t) 2c −2a (8.24a) 3a n=3 c n! dtn in the proper frame of reference of the charge is negligible compared to the ﬁrst two terms on the right-hand side of (8.6). For this sum to be negligible, it is suﬃcient that & & & n & &d u& 2a && dn+1 u && & & (n + 1) (8.24b) & & & dtn & , n = 2, 3, . . . . n+1 c dt Moreover, it can be shown that the nonlinear O(a)/mes terms are negligible compared to the ﬁrst two terms on the right-hand side of the proper-frame version of (8.6) if, in addition to the conditions in (8.24b), we also have that in the proper reference frame & & a && du && c (8.24c) c & dt & and

& & & & & du & a && d2 u && && && . c & dt2 & dt

(8.24d)

The conditions (8.24b)–(8.24d) are suﬃcient conditions for neglecting the terms in the proper-frame equation of motion beyond the radiation reaction

8.2 Cause and Elimination of the Pre-Acceleration

75

term. The conditions in (8.24b) and (8.24d) say that the fractional change in the ﬁrst and higher time derivatives of velocity of the charge in the proper frame is small during the time interval it takes light to traverse the charge distribution. The condition (8.24c) says that the velocity of the charge changes by a small fraction of the speed of light in the time interval light takes to traverse the charge distribution. The pre-acceleration solution (8.16) (with mes replacing m) behaves as exp[3cτ /(4a)] for τ < 0 and thus does not satisfy the condition that (8.24a) is negligible compared to the radiation reaction term because dn+1 3c dn exp[3cτ /(4a)] = exp[3cτ /(4a)]. dτ n+1 4a dτ n

(8.25)

(For the sake of argument, we are assuming that the conditions on the derivatives of the velocity function V for neglecting the O(a) terms in (8.6) are at least as strong as those on the derivatives of the proper-frame velocity in (8.24b)–(8.24d).) Moreover, the nonlinear O(a)/mes terms in (8.6) for this pre-acceleration solution have (exp[3cτ /(4a)])n dependence with n ≥ 2 and thus the nonlinear O(a)/mes terms cannot cancel the linear O(a)/mes terms in this pre-acceleration solution. Thus, one could initially conclude that the pre-acceleration solution in (8.16) may not be a valid solution to the equation of motion (8.6) for the charged insulator of radius a when the O(a)/mes terms are retained. Unfortunately, when the O(a)/mes terms are retained, the pre-acceleration (runaway solution for τ < 0) is not eliminated, just the time dependence of the pre-acceleration is altered. Speciﬁcally, the analyses of Herglotz [39] and Wildermuth [40] show that runaway solutions to the linearized, homogeneous form of our equation of motion (7.1) exist for all time, so that pre-acceleration exists for t < 0, regardless of how many linear higher order terms are included in the linearized equation of motion [41]. (These results of Herglotz and Wildermuth apply to the charged insulator even when mins = 0 as long as the sum of the bare mass and material mass of the insulator, M0 + mins , is less than zero. This condition is met as a → 0 because the bare mass has the negative value of M0 = −mes /3 and even when the mass of the insulator is not zero, the value of the sum mins − mes /3 → −mes /3 as a → 0.) The analyses of Herglotz and Wildermuth are approximate in that they neglect all O(a)/mes terms involving nonlinear products of the time derivatives of velocity in the proper-frame equation of motion (see Section 8.4). However, the analysis of motion of the two-charge (dumbbell) problem [42], although it neglects the self force of each individual charge, includes nonlinear terms and also exhibits the existence of runaway solutions. Thus, in general, the inclusion of higher order terms in the equation of motion fails to eliminate the pre-acceleration, whether or not the mass is kept at a ﬁxed value as a → 0. This conclusion is also conﬁrmed by the work of Bauer and D¨ urr [43] on the nonrelativistically rigid model of the extended electron.

76

8 Solutions to the Equation of Motion

8.2.1 Cause of the Pre-Acceleration The cause of the pre-acceleration can be found by examining the assumptions involved in the derivation of the equation of motion. In Chapters 2 through 5 and the Appendices, the equation of motion was obtained for the extended model of the electron as a spherical charged insulator of radius a. To simplify the discussion, concentrate on the force equation of motion (7.12a) in the proper frame of reference of the charged insulator Fext (t) = mu˙ −

e2 ¨ + O(a) . u 6π0 c3

(8.26)

As explained in Section 5.1, the rest mass, or coeﬃcient of the u˙ term in (8.26), is determined ultimately, not from the electromagnetic self force, but from the relativistic generalization of Newton’s second law of motion and the Einstein mass-energy relation. In particular, the rest mass must equal the total energy of formation of the charged insulator divided by c2 . ¨ and higher order reaction terms in the equation of motion (8.26) The u are determined from the derivation of the self electromagnetic force. This derivation, outlined in Appendix A, depends upon expanding the position, velocity, and acceleration of each element of charge at the retarded time (t = t − R (t )/c) in a Taylor series about the present time (t). For example, the velocity of the element of charge at r in the proper frame is expanded as

R (t ) R (t ) R2 (t ) ˙ , t) ¨ (r , t) = −u(r +u + · · · (8.27a) u(r , t ) = u r , t − c c 2c2 ˙ , t ), where the distance R (t ) has the Taylor series exand similarly for u(r pansion ˙ , t) R(t)R · u(r R (t ) = R(t) − + ··· . (8.27b) 2 2c These Taylor series expansions are valid provided the velocity function u(r , t ) is an analytic function of complex time t for |t − t| ≤ [R (t )/c]max .

(8.28)

˙ , t ) through (Analyticity of u(r , t ) implies the analyticity of R (t ) and u(r integration and diﬀerentiation, respectively.) For the self-force calculation in the proper frame of reference, R (t ) does not exceed a value of about 2a (assuming the velocity does not change rapidly between t and t; in other words, assuming the velocity change is a small fraction of the speed of light during the time it takes light to traverse the charge distribution — a condition implied by (8.24c)), and thus (8.28) can be rewritten as |t − t| ≤ Δta where

(8.29a)

8.2 Cause and Elimination of the Pre-Acceleration

77

2a . (8.29b) c Even if the magnitude of the velocity change (Δu) during the time 2a/c is a signiﬁcant fraction of the speed of light, (8.29b) can be replaced by Δta ≈

Δta ≈

1 2a c 1 − Δu/c

(8.29c)

which still approaches zero as a → 0 for Δu < c. Assume that the external force Fext (t) has at most a ﬁnite jump discontinuity 2 across t = 0 and that after the external force is applied at t = 0, the external force Fext (t), and the velocity of the charge u(r, t), is an analytic function of t in a Δta -wide strip of the complex t plane about the real t axis for t > 0. Then the Taylor series expansions hold for t > Δta , the equation of motion in (8.26) is valid for t > Δta , and with the O(a) terms neglected it implies that u(t) is an analytic function of time for t > Δta if and only if Fext (t) is an analytic function of time for t > Δta .3 However, since the external force and velocity are zero for t < 0, the velocity cannot be an analytic function of t for t in the interval of (8.29a) if t is less than Δta (and t ≥ 0). Also, note that because t ≤ t and u(r, t) with all its time derivatives are zero for t < 0, the left- and right-hand sides of the equations in (8.27) are equal for t < 0. In all then, the Taylor series expansions in (8.27) are invalid during (and only during) the time interval 0 ≤ t ≤ Δta .

(8.30)

Consequently, the following expression obtained by inserting equation (A.10) into (A.1) of Appendix A for the self electromagnetic force in the proper frame is not valid during this short transition time interval (8.30) in which the external force is ﬁrst applied ! % ˆ " r · u˙ 1 1 R ˆ · u) ˆ + u˙ ˙ R Fem (t) = + − 1 (R 2 2 2 4π0 R 2c R c charge

+ 2

3

ˆ ! 3R 8 c4

ˆ · u) ˙ − |u| ˙ (R 2

2

"

' ˆ · u) ˙ u˙ 3(R 2¨ u + + 3 + O(R) de de, u = 0. (8.31) 4c4 3c

A ﬁnite external force ensures that the external force is integrable in time and that the acceleration outside the transition interval in (8.30) is ﬁnite; see the last paragraph of Section 8.2.2. ˙ If a solution to the diﬀerential equation Fext (t) = mu(t) − [e2 /(6π0 c3 )]¨ u(t) (which holds for t > Δta ) exists in a complex neighborhood of the real time t, ˙ ¨ (t) exist in this complex neighborhood of real t, and thus u(t) is then u(t) and u an analytic function of time if Fext (t) is an analytic function of time at time t. If Fext (t) is not analytic at some time t, then u(t) cannot be analytic at time t because if it were the right-hand side of the diﬀerential equation would be analytic at time t while the left-hand side would not.

78

8 Solutions to the Equation of Motion

One can see directly from equation (A.2) of Appendix A how the integral in (8.31) should be modiﬁed for t < R (t )/c. Speciﬁcally, for t < R (t )/c, the ˙ , t ) are identically zero so that dE(r, t) in (A.2) functions u(r , t ) and u(r ˆ reduces to de R /(4π0 R2 ). Thus, a simple modiﬁcation to the integrand of (8.31), shown in the following equation (8.32), produces an expression for the self electromagnetic force that is valid for all time in the proper frame

( ˆ % ˆ ˆ 1 R (t ) R 1 R R + h t − − + 2 Fem (t) = 4π0 R2 c R2 R2 2c R charge

! " " ˆ ! r · u˙ 2 2 ˆ · u) ˆ · u) ˆ + u˙ + 3 R (R ˙ ˙ ˙ − 1 ( R R − | u| · c2 8 c4 )' ˆ · u) ˙ u˙ 3(R 2¨ u + + 3 + O(R) de de, u = 0 (8.32) 4 4c 3c

where h(t) in (8.32) is the unit step function deﬁned by 0, t<0 h(t) = 1, t≥0.

(8.33)

Although the step function appearing in (8.32) represents a minor modiﬁcation, it prevents the closed-form evaluation of the double integration in (8.32) during the time interval (8.30). Nonetheless, for t < 0, the step function h(t − R (t )/c) = 0 and R (t ) = R. Therefore, since R/R2 de de = 0, charge the value of the integral in (8.32) is zero for t < 0, and for t > Δta , the integral yields the usual expression (A.11) for the self electromagnetic force in the proper frame. During the transition time interval in (8.30) after the ﬁnite external force is ﬁrst applied, there appears to be no simple way to evaluate the double integral in (8.32) to determine the self electromagnetic force because the time dependence of u(r , t ), and thus R (t ) in h[t − R (t )/c], is unknown a priori.4 Nonetheless, the diﬀerence between the correct expression for the self electromagnetic force in (8.32) and the original expression in (8.31) implies that a transition force must be added to the original self electromagnetic force in (A.11) during the time interval in (8.30). In the following section, we show that such a transition force can be chosen to eliminate the pre-acceleration from the equation of motion. Abraham also realized that the traditional series representation of the self electromagnetic force became invalid for “discontinuous movements” of the charge. In [3, sec. 23] he states, “These two forces [electromagnetic mass term 4

It was argued in the ﬁrst edition of the book that the self electromagnetic force in (8.32) approached zero as O(t2 ) from positive values of t, but this argument assumed that u(r , t ) was slowly varying in the transition interval (8.30), and this is not necessarily the case.

8.2 Cause and Elimination of the Pre-Acceleration

79

plus radiation reaction] are basically nothing other than the ﬁrst two terms of a progression which increases in accordance with increasing powers of the electron’s radius a. . . . Because the internal force is determined by the velocity and acceleration existing in a ﬁnite interval preceding the aﬀected point in time, such a progression is always possible when the movement is continuous and its velocity is less than the speed of light. . . . The series will converge more poorly the closer the movement approaches a discontinuous movement and the velocity approaches the speed of light . . . . It fails completely for discontinuous movements. . . . Here, other methods must be employed when computing the internal force.” Abraham goes on to derive the radiated energy and momentum of a charged sphere with discontinuous velocity [3, sec. 25], [44]. He also derives Sommerfeld’s general integral formulas for the internal electromagnetic force [45]. Neither he nor Sommerfeld, however, evaluates or interprets these general integrals except to show they yield a null result for a charged sphere moving with constant velocity. Schott [46], [16, p. 283] also concludes that “the approximation [used to obtain the Lorentz-Abraham equation of motion] fails during an interval of time, which is comparable with the time required by an electromagnetic wave to pass across the electron and includes the instant at which the discontinuity occurs.” More recently, Valentini [47] observes that “the usual derivations of the Lorentz-[Abraham-]Dirac equation are only valid at times such that [the position of and force applied to the particle] are analytic functions [of time],” and that nonanalyticity of these functions is responsible for the noncausal pre-acceleration in the solution to the Lorentz-Abraham-Dirac equation of motion. 8.2.2 Elimination of the Pre-Acceleration We have shown that (8.32) rather than (8.31) is the formal expression for the self electromagnetic force on the spherical shell of charge that holds for all time t in the proper inertial reference frame and reduces to (8.31) for t > Δta and t < 0 under the assumptions that the externally applied force is zero for t < 0, has at most a ﬁnite jump discontinuity across t = 0, and is an analytic function of complex t about the real t axis for t > 0. As pointed out in the previous subsection, however, the double integration over the charge in (8.32) cannot, in general, be evaluated in closed form during the time interval 0 ≤ t ≤ Δta because of the presence of the step function h(t − R (t )/c) in the integrand of (8.32) and because of the unknown behavior of R (t ) during this time interval. Nonetheless, (8.32) can still be re-expressed as a diﬀerential equation that leads to an equation of motion free of pre-acceleration. To do this, begin with the evaluation of (8.32) for t > Δta (or t < 0). Then (8.32) equals (8.31) which is simply the original self force evaluated in (A.11), that is Fem (t) = −

e2 e2 ¨ + O(a) , u˙ + u 2 6π0 ac 6π0 c3

t > Δta , t < 0 .

(8.34)

80

8 Solutions to the Equation of Motion

Although (8.32) cannot be evaluated in closed form in the interval 0 ≤ t ≤ Δta , we can see that the step function in (8.32) changes the coeﬃcients ¨ in the equation of motion during this time interval. Thus, we can of u˙ and u formally express the self electromagnetic force for t ≥ 0 as Fem (t) = −

e2 e2 ˙ η (t) u + η (t)¨ u + O(a) 1 6π0 ac2 6π0 c3 2

(8.35)

where the functions η1 (t) and η2 (t) equal zero at t = 0, equal 1 for t > Δta and t < 0, and have some unknown, possibly rapid variation that depends on a and the behavior of u(r , t ) (and thus R (t )) in the interval 0 ≤ t ≤ Δta . Following the procedure used in Section 5.1, we assume the relativistic version of Newton’s second law of motion requires that the external force applied to a charged particle should equal, apart from the radiation reaction and the forces of order a, the time derivative of momentum of the particle. That is, ˙ Fext (t) = mu(t) − η2 (t)

e2 ¨ (t) + O(a) u 6π0 c3

(8.36)

where m = mes + mins is the rest mass of the particle (charge plus insulator) and the required bare mass term is no longer (mes − mem )u˙ but ˙ 5 Since the functional time dependence of η2 (t) is unknown, [mes −η1 (t)mem ]u. 5

The corrected equation of motion (8.36) can thus be re-expressed as Fext (t) + Fem (t) = [M0 + mins + (1 − η1 (t)) mem ]u˙

(8.37)

where, as in Chapter 5, M0 = mes − mem . If one postulates a priori an equation of motion for a classical charged particle with speciﬁed rest mass m given in the proper frame by Fext (t) + Fem (t) = (M0 + mins )u˙ = M u˙ without the (1 − η1 (t)) mem u˙ term, an initial-value solution free of pre-acceleration and runaway behavior does not generally exist for small values of a [43]. In other words, for small enough a and a given rest mass m, causality is violated if one assumes in an arbitrary inertial frame that the equation of motion has the form d(γu) Fext (t) + ρ(r, t)[E(r, t) + u(r, t) × B(r, t)]dV = M (8.38) dt charge

regardless of the value chosen for the constant “material mass” M or the velocity u(r, t) [38], [49]. This result does not imply that Maxwell’s equations with Lorentz forces are inconsistent, only that the postulated equation of motion in (8.38) (which does not include the (1 − η1 (t)) mem u˙ term) is ﬂawed for the classical model of an accelerating charged particle with a given charge e and rest mass m. The more fundamental equation of motion that is used in (8.36) and its relativistic generalization (8.43) equates the sum of the externally applied force and the radiation reaction part of the self force to the rest mass of the particle times the acceleration. As will be shown, this more fundamental equation of motion, unlike (8.38), allows for a well-behaved causal initial-value solution free of runaway motion.

8.2 Cause and Elimination of the Pre-Acceleration

81

divide the equation in (8.36) by η2 (t) to get6 Fext (t) m e2 ¨ (t) + O(a) . ˙ u = u(t) − η2 (t) η2 (t) 6π0 c3

(8.39)

We can rewrite this equation as ˙ − Fext (t) + fa (t) = mu(t)

where fa (t) =

e2 ¨ (t) + O(a) u 6π0 c3

0 , ˙ [Fext (t) − mu(t)] [1/η2 (t) − 1] ,

t<0 t ≥ 0.

(8.40)

(8.41)

Although the exact time dependence of the transition force fa (t) is unknown, it vanishes outside the interval given in (8.30) and it may be a complicated function of time whose contribution to the equation of motion in (8.40) does not necessarily approach zero as a → 0. In other words, fa (t) may contain delta functions and their time derivatives as a → 0. (For example, if the actual u(t) has a jump across t = Δta as a → 0 and Fext (t) is ﬁnite, equation (8.40) implies that fa (t) + O(a) contains a doublet function across t = Δta .) In brief, a causal equation of motion with the correct rest mass of the charged particle requires a transition force fa (t) to remove the pre-acceleration and a modiﬁed bare mass term [mes − η1 (t)mem ]u˙ to produce the correct rest mass. The modiﬁed proper-frame equation of motion in (8.40) can be rewritten Fext (t) + fa (t) O(a) ¨+ = u˙ − τe u m m

(8.42)

where the time constant τe is the same as in (8.7). In an arbitrary inertial reference frame, and in four-vector notation, (8.42) generalizes to

2 i i j O(a) + fai Fext dui d u i duj du + = c + u − τ (8.43) e mc2 ds ds2 ds ds m with the four-vector force given by fai = γ (fa , fa · u/c) 6

(8.44)

In the ﬁrst edition of this book, we left the corrected equation of motion in the form of (8.36), in which the function η2 (t) was assumed to vary monotonically from a value of zero at t = 0 to a value of unity at t = Δta . Although this simpliﬁed approximation to η2 (t) eliminated the pre-acceleration, a similar approximation to the corrected equation of motion to eliminate pre-deceleration can lead to a violation of energy conservation, as shown by Baylis and Huschilt [48]. The general form (8.40) of the corrected equation of motion allows conservation of momentum and energy while eliminating both pre-acceleration and pre-deceleration for the extended charged insulator model, as demonstrated in Section 8.2.5.

82

8 Solutions to the Equation of Motion

such that fai is zero outside the interval (0 ≤ s ≤ Δsa ≈ cΔta ) and fai ui = 0. (Note that Δsa ≈ cΔta even if the initial velocity is not zero because Δsa is an invariant and it equals cΔta in the proper inertial reference frame.) The modiﬁed equation of motion (8.43) can be rewritten in three-vector notation from (7.12) as d d O(a) γ2 Fext + fa d(γu) γ4 ˙ 2 + 2 (u · u) ˙ 2 u + γ (γu) − 2 |u| = − τe m dt dt dt c c m (8.45a) and

dγ γ4 d O(a) dγ γ2 (Fext + fa ) · u 2 2 ˙ + 2 (u · u) ˙ − τe γ − 2 |u| + = mc2 dt dt dt c c m (8.45b) 2 ˙ in which we have used the relations γd(γu)/dt = γ 2 u˙ + γ 4 (u · u)u/c and 4 2 ˙ . γdγ/dt = γ u · u/c An extremely simple way to arrive at (8.43) (or (8.45)) is to observe that outside the transition interval this equation holds rigorously without the transition function (under the assumption that the sum of the external force and the radiation reaction part of the self force equals the rest mass times the relativistic acceleration). Within the transition interval, dui /ds − τe c[d2 ui /ds2 + i ui (duj /ds)(duj /ds)] + O(a)/m − Fext /(mc2 ) has to equal some unknown function of time that can simply be denoted by fai /(mc2 ). A formal solution to (8.45) with the O(a)/m terms neglected can be ob˙ tained for u(t) by replacing Fext in (8.21) with (Fext + fa ) to get

1 ˙ u(t)= mτe γ 3 (t)

∞

γ(t ) Fext (t ) + fa (t )

t

− [(Fext (t ) + fa (t )) · u(t )] u(t )/c

2

(8.46a) !

exp − (1/τe )

t

" γ −1 (t )dt dt

t

or, equivalently, with the change of integration variables dt /γ = dτ and dt /γ = dτ 1 ˙ )= u(τ mτe γ 3 (τ )

∞

# γ 2 (τ ) Fext (τ ) + fa (τ )

(8.46b)

τ

$ − [(Fext (τ ) + fa (τ )) · u(τ )] u(τ )/c2 exp[−(τ − τ )/τe ]dτ

which has to be zero for t < 0 under the assumption that the exact classical solution is causal and the externally applied force Fext is zero for t < 0. As is customarily done with u and Fext , the same symbol is used for fa in (8.42)– (8.46) independently of its inertial reference frame or its dependent variable s, t, or τ .

8.2 Cause and Elimination of the Pre-Acceleration

83

The necessity of a transition force fa (t), which contributes only during the short time it takes light to travel across the charged sphere, can be understood physically by considering two diﬀerential elements of charge at either end of the charge distribution. These two elements are at rest separated by a distance 2a. When the external force is ﬁrst applied, each of these charge elements accelerates and radiates. However, each element of charge does not experience the radiation from the other until approximately the time it takes light to travel between them . Thus, there will be a time delay in the radiation reaction force of about 2a/c between these two elements of charge separated by 2a (assuming here that the velocity of the charge does not change by an appreciable fraction of c during this time interval). For the other combinations of charge elements separated by a distance less than 2a, the time delay of the radiation will be proportionately less. The double integration over the entire sphere of charge elements de and de produces a continuous addition of radiation forces with time delays varying from zero to about 2a/c. Moreover, the trailing end of the charge ﬁrst experiences the change in ﬁeld from the leading end a short time before the leading end of the charge ﬁrst experiences the change in ﬁeld from the trailing end. In all, the transition force fa (t) appears between the time the external force is ﬁrst applied and the time Δta ≈ 2a/c after which the self electromagnetic force can be expressed entirely in terms of the present velocity and its time derivatives. The functional time dependence of this transition force must also account for the addition of the bare mass in order to maintain causality in the solution to the equation of motion. The transition force fa (t) in the equation of motion allows solutions to the equation of motion that satisfy initial conditions on velocity and that are free of pre-acceleration. Although the exact dependence on time of the function fa (t) is unknown, it vanishes outside the interval in (8.30), that is fa (t) = 0 ,

t ∈ [0, Δta ].

(8.47)

The requirement that the function fa (t) eliminates the pre-acceleration further restricts the time dependence of fa (t). The necessary and suﬃcient condition for there to be no pre-acceleration can be expressed formally by demanding ˙ that u(t) = 0 for t < 0 in the solution (8.46) to obtain from (8.46b) Δta * + γ 2 (τ ) fa (τ ) − [fa (τ ) · u(τ )] u(τ ) e−τ /τe dτ 0

∞ =−

* + γ 2 (τ ) Fext (τ ) − [Fext (τ ) · u(τ )] u(τ ) e−τ /τe dτ .

(8.48)

0

The important feature of (8.48) that allows this condition to be satisﬁed is that the τ dependence (eτ /τe ) has canceled to leave the left- and right-hand sides of (8.48) independent of τ (and thus independent of t). Moreover, if

84

8 Solutions to the Equation of Motion

u(τ ) and Fext (τ ) are expandable in a power series for τ > 0, the function γ 2 (τ ){Fext (τ ) − [Fext (τ ) · u(τ )]u(τ )} in the integrand on the right-hand side of (8.48) can be expanded in a power series about τ = 0+ . Term by term integration of this power series then shows that the right-hand side of (8.48) can be expressed solely in terms of the values of the external force and velocity and their time derivatives at τ = 0+ . This re-expression of the right-hand side of (8.48) is philosophically appealing because it removes the apparent dependence of the right-hand side of (8.48) on values of the external force later than τ = 0+ (t = 0+ ) and thus later than τ = Δta after which time fa (τ ) is zero. To determine a more explicit form of the condition in (8.48) that applies to rectilinear motion, rewrite (8.43) for rectilinear motion in the form of (8.6) by means of the change of variables deﬁned at the beginning of Chapter 8 to get Fext (τ ) + fa (τ ) O(a) . (8.49) = V (τ ) − τe V (τ ) + m m As usual, Fext (τ ), and now fa (τ ), are compact notations for Fext [t(τ )] and fa [t(τ )], and it is assumed that τ = 0 when t = 0. As in (8.13) and (8.16), the solution to (8.49) for all τ under the asymptotic condition in (8.12) and with the O(a)/m terms neglected is given by 1 V (τ ) = mτe

1 V(τ ) = m

∞ [Fext (τ ) + fa (τ )] exp[−(τ − τ )/τe ]dτ

(8.50a)

τ

∞ [Fext (τ ) + fa (τ )] exp[−(τ − τ )/τe ]dτ τ

+

1 m

τ

[Fext (τ ) + fa (τ )]dτ

0

= τe V (τ ) +

1 m

τ

[Fext (τ ) + fa (τ )]dτ

(8.50b)

0

where in (8.50b) it is assumed that the velocity is zero before the external force is applied. Since Fext (τ ) and velocity are zero for τ < 0, the velocity and acceleration functions must be zero for τ < 0. Thus, (8.50) imply ∞ [Fext (τ ) + fa (τ )]e−τ /τe dτ = 0 0

or

(8.51a)

8.2 Cause and Elimination of the Pre-Acceleration Δta ∞ −τ /τe fa (τ )e dτ = − Fext (τ )e−τ /τe dτ 0

85

(8.51b)

0

a result that can also be gotten directly from (8.48) by noting that the integrands in (8.48) contain the factor (1 − u2 /c2 ) = 1/γ 2 for rectilinear motion. The integrals of fa (τ ) have the ﬁnite limits given in (8.48) and (8.51b) because fa (τ ) is zero outside the interval 0 ≤ τ ≤ Δta . If there are more than one nonanalytic point in the otherwise analytic external force function Fext (τ ), for example, at τ = τ1 = 0 and at τ = τ2 , rewrite Fext (τ ) as Fext (τ ) = F1 (τ ) + h(τ − τ2 )[Fext (τ ) − F1 (τ )]

(8.52)

in which h(τ ) is the unit step function deﬁned in (8.33). The function F1 (τ ) is deﬁned as follows: F1 (τ ) = Fext (τ ) for τ < τ2 (Fext (τ ) = 0, τ < 0) and, in addition, F1 (τ ) is the analytic continuation in the complex τ plane about the real τ axis of Fext (τ ) from τ < τ2 to τ ≥ τ2 . Thus, F1 (τ ) is an analytic function of complex τ about the real τ axis for τ > τ1 = 0. Moreover, the function in the square bracket, Fext (τ ) − F1 (τ ), is an analytic function of complex τ about the real τ axis for all τ > τ2 . There will now be two transition functions, that is, fa (τ ) = fa1 (τ )+fa2 (τ ), where the ﬁrst, fa1 (τ ), is nonzero only in the interval [0, Δta ] and the second, fa2 (τ ), is nonzero only in the interval [τ2 , τ2 + Δta ]. Then the solutions for V (τ ) and V(τ ) in (8.50) become 1 V (τ ) = mτe

+

1 mτe

∞

∞ [F1 (τ ) + fa1 (τ )] exp[−(τ − τ )/τe ]dτ

(8.53a)

τ

h(τ − τ2 )[Fext (τ ) − F1 (τ ) + fa2 (τ )] exp[−(τ − τ )/τe ]dτ

τ

1 V(τ ) = τe V (τ ) + m

τ

[Fext (τ ) + fa (τ )]dτ ,

V(0) = 0 .

(8.53b)

0

For there to be no pre-acceleration before τ = τ1 = 0 and no pre-acceleration (or pre-deceleration) before τ = τ2 , the transition functions fa1 (τ ) and fa2 (τ ) must satisfy ∞ [F1 (τ ) + fa1 (τ )]e−τ /τe dτ = 0 (8.54a) 0

and

∞ [Fext (τ ) − F1 (τ ) + fa2 (τ )]e−τ /τe dτ = 0 τ2

(8.54b)

86

8 Solutions to the Equation of Motion

or, equivalently Δta ∞ −τ /τe fa1 (τ )e dτ = − F1 (τ )e−τ /τe dτ 0

(8.55a)

0

and Δta ∞ −τ /τe fa2 (τ + τ2 )e dτ = − [Fext (τ + τ2 ) − F1 (τ + τ2 )]e−τ /τe dτ. (8.55b) 0

0

With the help of (8.54) or (8.55), the solution for written for τ outside the transition intervals as ⎧ 0 , ⎪ ⎪ ∞ ⎪ ⎪ 1 ⎨ F1 (τ ) exp[−(τ − τ )/τe ]dτ , V (τ ) = τ mτe ⎪ ∞ ⎪ ⎪ ⎪ ⎩ Fext (τ ) exp[−(τ − τ )/τe ]dτ , τ

V (τ ) in (8.53a) can be τ <0 Δta < τ < τ2

(8.56)

τ2 + Δta < τ .

For N nonanalytic points in time τn of the otherwise analytic external force function Fext (τ ), the ﬁrst occurring at τ = τ1 = 0 when the externally applied force is ﬁrst applied to the charged sphere, there are transition functions fan (τ ) nonzero only in the interval [τn , τn + Δta ] such that -N fa (τ ) = n=1 fan (τ ). The external force can be expressed in terms of N functions [F1 (τ ), F2 (τ ), · · · , FN (τ )] such that Fn (τ ) is the analytic continuation of Fext (τ ) about the real τ axis from τn < τ < τn+1 to τ ≥ τn+1 for n = 1, 2, · · · , N − 1, with FN (τ ) = Fext (τ ) for τN < τ < ∞, and F1 (τ ) = Fext (τ ) = 0 for τ < 0: Fext (τ ) = F1 (τ ) + h(τ − τ2 )[F2 (τ ) − F1 (τ )] + · · · + h(τ − τN )[FN (τ ) − FN −1 (τ )].

(8.57)

Therefore, F1 (τ ) is an analytic function of complex τ about the real τ axis for τ > τ1 = 0, and each of the functions in the square brackets, Fn+1 (τ ) − Fn (τ ), n = 1, 2, · · · , N − 1, is an analytic function of complex τ about the real τ axis for τ > τn . The acceleration function V (τ ) satisﬁes the diﬀerential equation ( ) N 1 O(a) Fext (τ ) + fan (τ ) = V (τ ) − τe V (τ ) + (8.58) m m n=1 which has the exact solution (neglecting the O(a)/m terms) given by

8.2 Cause and Elimination of the Pre-Acceleration

1 V (τ ) = mτe

+

1 mτe

.. . 1 + mτe

∞

∞ [F1 (τ ) + fa1 (τ )] exp[−(τ − τ )/τe ]dτ

87

(8.59a)

τ

h(τ − τ2 )[F2 (τ ) − F1 (τ ) + fa2 (τ )] exp[−(τ − τ )/τe ]dτ

τ

∞

h(τ − τN )[FN (τ ) − FN −1 (τ ) + faN (τ )] exp[−(τ − τ )/τe ]dτ

τ

and 1 V(τ ) = τe V (τ ) + m

τ (

Fext (τ ) +

N

)

fan (τ ) dτ , V(0) = 0

(8.59b)

n=1

0

with the transition functions fan (τ ) eliminating all pre-acceleration and predeceleration by satisfying the conditions Δta ∞ fan (τ + τn )e−τ /τe dτ = − [Fn (τ + τn )−Fn−1 (τ + τn )]e−τ /τe dτ , 0

0

n = 1, 2, · · · , N

(8.60)

with τ1 = 0 and F0 (τ ) ≡ 0. The conditions (8.60) on the transition functions fan (τ ) allow V (τ ) in (8.59a) to be expressed outside the transition intervals [τn , τn + Δta ] as simply ⎧ , τ < τ1 = 0 ⎨0 1 ∞ V (τ ) = mτe ⎩ Fn (τ ) exp[−(τ − τ )/τe ]dτ , τn + Δta < τ < τn+1 , τ

n = 1, 2, · · · , N

(8.61)

where τN +1 = ∞. This result says that the acceleration function V (τ ) is determined at a time τ outside the transition intervals by the values of the externally applied force at the time τ and its analytic continuation to τ = ∞. In each domain τn + Δta < τ < τn+1 , the acceleration function V (τ ) is the particular solution to the diﬀerential equation of motion Fn (τ ) = V (τ ) − τe V (τ ) , m

τn < τ < ∞

(8.62)

that converges as τ → ∞. The transition functions fan (τ ), which are nonzero only in the transition intervals [τn , τn + Δta ], may contain delta functions and their derivatives as

88

8 Solutions to the Equation of Motion

Δta → 0. That is, even as a → 0, the acceleration function V (τ ) may produce a ﬁnite jump in the velocity function V(τ ) across the transition intervals. Since V (τ ) in (8.61) is not given for τ in the transition intervals, (8.61) cannot be used to determine these possible jumps in velocity across the transition intervals. The diﬀerential equation for V (τ ) valid for all τ is given by (8.58). Its solution for V (τ ) and V(τ ) given in (8.59) can also be expressed as V (τ ) =

N

Vn (τ )

(8.63a)

n=1

V(τ ) =

N

Vn (τ ) ,

Vn (τn ) = 0

(8.63b)

n=1

where Vn (τ ) =

1 mτe

∞

h(τ − τn )[Fn (τ ) − Fn−1 (τ ) + fan (τ )] exp[−(τ − τ )/τe ]dτ

τ

(8.63c) and Vn (τ ) =

τe Vn (τ )

1 + m

τ

h(τ − τn )[Fn (τ ) − Fn−1 (τ ) + fan (τ )]dτ (8.63d)

τn

for all τ , so that Vn (τ ) satisﬁes h(τ −τn )[Fn (τ )−Fn−1 (τ )+fan (τ )]/m = Vn (τ ) − τe Vn (τ ), −∞ < τ < ∞ (8.63e) with Vn (τn ) = 0. The jumps in the acceleration functions across the nth transition interval [τn , τn +Δta ] are independent of fan (τ ), whereas the jumps in the velocity functions are determined by fan (τ ). These jumps are given by ΔV = ΔVn and ΔV = ΔVn . Note, however, that, in general, V (τn ) = Vn (τn ) = 0, V(τn ) = Vn (τn ) = 0, V (τn + Δta ) = Vn (τn + Δta ) = ΔVn , and V(τn + Δta ) = Vn (τn + Δta ) = ΔVn . The modiﬁed equation of motion (8.43) or (8.45), and its rectilinear version (8.58), still admit a homogenous runaway solution. However, this runaway solution is eliminated from the modiﬁed equation of motion by the asymptotic condition in (8.12) without introducing noncausal motion into the solution. Although the solution to the modiﬁed equation of motion (8.45) is free of pre-acceleration, it may be bothersome that for τ > τn + Δta the solution in (8.61) to the modiﬁed (and original) equation of motion depends on the values of the external force at all future times. This result becomes understandable if it is remembered that (8.61) is the solution to an equation of motion obtained under the restriction that the externally applied force function Fn (τ ) be an analytic function of time about the real τ axis for all τ > τn , because the

8.2 Cause and Elimination of the Pre-Acceleration

89

values of an analytic function on an interval of a singly connected domain of analyticity determine uniquely the function over the rest of the domain. For example, assume that for τ > τn the external force Fn (τ ) in (8.61) can be expanded in a power series about τ to recast (8.61) in the form V (τ ) =

∞

dk Fn (τ ) 1 (τe )k , m dτ k

τn + Δta < τ < τn+1

(8.64)

k=0

which simply states that the acceleration at any one time (τn + Δta < τ < τn+1 ) depends on the time derivatives of the applied force as well as the applied force itself at that time. (Note that (8.64) is not a valid representation for V (τ ) in the transition interval τn < τ < τn + Δta containing the transition force in addition to the externally applied force.) If the restriction that the external force Fn (τ ) be an analytic function of τ for all τ > τn is ignored, and Fn (τ ) is allowed to attain a strong enough inﬁnite singularity at some future point in time, as in the case of the charged sphere being attracted to the center of a Coulomb ﬁeld (1/r2 singularity), the integration in (8.61) may not converge for all values of τ before the sphere reaches the singularity [50]. 8.2.3 Determination of the Transition Force for Rectilinear Motion Although the exact variation with τ of each fan (τ ) is unknown, they can be represented operationally from equations (8.63) in terms of delta functions as the radius a of the charge approaches zero so that Δta → 0 and the mass m is renormalized to a prescribed value. From (8.63c)–(8.63d) we see that Vn (τ ) = 0 for τ < τn and it is a diﬀerentiable function of τ for τ > τn + Δta , that is, for τ ≥ τn+ as Δta → 0, where τn+ indicates a value inﬁnitesimally larger than τn . Thus, we can represent Vn (τ ) for all τ as Vn (τ ) = h(τ − τn+ )Vn+ (τ )

(8.65)

with h(τ ) denoting the unit step function and the Vn+ (τ ) on the right-hand side of this equation is a diﬀerentiable function of τ equal to Vn (τ ) for τ ≥ τn+ . Diﬀerentiating this equation produces Vn (τ ) = δ(τ − τn+ )Vn+ (τ ) + h(τ − τn+ )Vn+ (τ )

(8.66a)

Vn (τ ) = δ (τ − τn+ )Vn+ (τ ) + 2δ(τ − τn+ )Vn+ (τ ) + h(τ − τn+ )Vn+ (τ ) (8.66b) where δ(τ ) is the Dirac delta function. Combining Vn and Vn , we have Vn − τe Vn = δ(τ − τn+ )[Vn (τn+ ) − τe Vn (τn+ )] − τe δ (τ − τn+ )Vn (τn+ ) + h(τ − τn+ )[Fn (τ ) − Fn−1 (τ )]/m (8.67)

90

8 Solutions to the Equation of Motion

in which we have used the equation of motion (8.63e) in the domain τ ≥ τn+ , namely, [Fn (τ ) − Fn−1 (τ )]/m = Vn (τ ) − τe Vn (τ ), as well as the operational relationships δ(τ − τn+ )Vn+ (τ ) = δ(τ − τn+ )Vn (τn+ ) and δ (τ − τn+ )Vn+ (τ ) = δ (τ − τn+ )Vn (τn+ ) − δ(τ − τn+ )Vn (τn+ ). Comparing (8.67) with the equation of motion (8.63e) shows that the transition force can be expressed as fan (τ ) = [ΔVn − τe ΔVn ] δ(τ − τn+ ) − τe ΔVn δ (τ − τn+ ) m

(8.68)

where ΔVn has replaced Vn (τn+ ), and ΔVn has replaced Vn (τn+ ). Since fan (τ ) alone determines ΔVn , we are free to decide the value of ΔVn in (8.68). However, ΔVn is determined solely by the externally applied force and is independent of fan (τ ). Thus, it is a parameter whose value cannot be changed in (8.68). Choosing ΔVn = 0 leaves only the delta function in (8.68) and makes the velocity function continuous. Choosing ΔVn = τe ΔVn leaves only the doublet function in (8.68) and produces a jump in velocity approximately equal to the change in velocity produced by the original pre-acceleration or in the original equation of motion (see Sections 8.2.4 and 8.2.5). We can verify that the transition functions in (8.68) remove the preacceleration (or pre-deceleration) by conﬁrming that they satisfy the conditions in (8.60). From (8.68) we see that Δta fan (τ + τn )e−τ /τe dτ = −mτe ΔVn

(8.69)

0

and from (8.63c) Vn (τn+ )

=

ΔVn

1 = mτe

1 = mτe

∞ [Fn (τ ) − Fn−1 (τ )] exp[−(τ − τn+ )/τe ]dτ + τn

∞ [Fn (τ + τn ) − Fn−1 (τ + τn )]e−τ /τe dτ

(8.70)

0

so that the conditions in (8.60) are satisﬁed and the pre-acceleration (or predeceleration) is eliminated. Inserting fan (τ ) from (8.68) into (8.63c) and (8.63d), we can write as a → 0 Vn (τ )

Vn (τ ) =

1 = h(τ − τn ) mτe

τe Vn (τ )

∞ [Fn (τ + τ ) − Fn−1 (τ + τ )]e−τ /τe dτ

(8.71a)

0

τ 1 + h(τ − τn ) ΔVn − τe ΔVn + [Fn (τ ) − Fn−1 (τ )]dτ m τn

(8.71b)

8.2 Cause and Elimination of the Pre-Acceleration

91

which when summed in (8.63a) and (8.63b) yield ⎧ 0 , τ < τ1 = 0 1 ⎨ ∞ V (τ ) = −τ /τe dτ , τn+ < τ < τn+1 , mτe ⎩ Fn (τ + τ )e 0

n = 1, 2, · · · , N

V(τ ) = τe V (τ ) +

N

(8.72a)

h(τ − τn )(ΔVn −

τe ΔVn )

n=1

or by substituting

ΔVn

1 + m

τ

Fext (τ )dτ (8.72b)

0

from (8.70) and

Vn (τ )

from (8.71a) into (8.63a)

∞ 1 h(τ − τn ) ΔVn + [Fn (τ + τ ) − Fn (τ + τn ) V(τ )= m n=1 N

0

−Fn−1 (τ + τ ) + Fn−1 (τ + τn )]e−τ

/τe

dτ

+

1 m

τ

(8.72c)

Fext (τ )dτ

0

expressions that are free of pre-acceleration and pre-deceleration. The equations in (8.71) and (8.72) satisfy the jump conditions ΔVn

=V

(τn+ )−V (τn )

1 = mτe

∞ [Fn (τ +τn )−Fn−1 (τ +τn )]e−τ /τe dτ (8.73a) 0

ΔVn = V(τn+ ) − V(τn ) .

(8.73b)

Although the jumps in the acceleration function are determined by the integrations of the externally applied force in (8.73a), the jumps in the velocity function in (8.73b) remain undetermined. The drawback of the corrected equation of motion, as discussed in Section 8.2.2, is that the exact time dependences of the transition functions fan (τ ) required by causality at each nonanalytic point in time τn of Fext (τ ) cannot be determined before the velocity of the charge is known. We have, however, been able to reduce this ambiguity in the fan (τ ) to the jumps ΔVn in the velocity function across the transition intervals. In Section 8.2.5, it is shown that reasonable values for the ΔVn can be chosen to conserve momentum and energy while maintaining a non-negative value for the energy radiated during the transition intervals. 8.2.4 Motion of Charge in a Uniform Electric Field for a Finite Time Assume a uniform electrostatic ﬁeld E0 is applied to a charge e with mass m initially at rest at t1 = τ1 = 0 and that the electric ﬁeld is turned oﬀ at time t = t2 (τ = τ2 ). For example, the charge could be accelerated between

92

8 Solutions to the Equation of Motion

two inﬁnitesimally thin plates of a parallel-plate capacitor charged to produce the electric ﬁeld E0 . It could be released at time t = 0 from one plate of the capacitor and leave through a small hole in the second plate at time t = t2 . With the O(a)/m terms neglected, the acceleration and velocity as a function of time can be found for the original equation of motion from (8.13a) and (8.16) (or from (8.59) without the transition forces) as ⎧ 1 − e−τ2 /τe eτ /τe , τ ≤ 0 ⎪ ⎪ ⎨ eE0 (8.74a) Vpre (τ ) = 1 − e(τ −τ2 )/τe , 0 ≤ τ ≤ τ2 m ⎪ ⎪ ⎩ 0 , τ2 ≤ τ ⎧ −τ2 /τe eτ /τe , τ ≤ 0 ⎪ ⎪ τe 1 − e eE0 ⎨ Vpre (τ ) = (8.74b) τe 1 − e(τ −τ2 )/τe + τ , 0 ≤ τ ≤ τ2 m ⎪ ⎪ ⎩ τ2 , τ2 ≤ τ where the subscripts “pre” indicate the solution to the original equation of motion that exhibits pre-acceleration and pre-deceleration. Note that the change in the velocity function due to the pre-acceleration before τ = 0 and the predeceleration before τ = τ2 is approximately equal to eE0 τe /m for τ2 τe . Also, the pre-acceleration involves τ2 and thus anticipates when the externally applied force turns oﬀ (at τ = τ2 ) as well as when it is ﬁrst applied (at τ = 0). The solution to the modiﬁed equation of motion that is free of noncausal pre-acceleration and pre-deceleration can be found by evaluating (8.72) for N = 2 to get (as a → 0 with renormalized mass m) ⎧ 0, τ <0 ⎪ ⎪ ⎨ eE 0 1 , 0+ < τ < τ2 V (τ ) = (8.75a) m ⎪ ⎪ ⎩ + 0 , τ2 < τ ⎧ 0 , τ <0 ⎪ ⎪ ⎨ (8.75b) V(τ ) = ΔV1 + eE0 τ /m , 0+ < τ < τ2 ⎪ ⎪ ⎩ + ΔV21 + eE0 τ2 /m , τ2 < τ with ΔV21 = ΔV2 + ΔV1 . Except for the homogeneous exponential (runaway) solution in (8.74) and the discontinuities across the transition intervals at τ = 0 and τ = τ2 in (8.75), the solutions in both (8.74) and (8.75) are identical to the solution one obtains by solving the equation of motion without the radiation reaction terms, that is, by solving the relativistic version of Newton’s second law of motion. This behavior, which depends uniquely on the external force being constant while it is applied, was discussed in Section 7.1 on hyperbolic (relativistically uniform) motion. We can make use of this behavior of the motion of a charge in a parallelplate capacitor to solve the rectilinear version of (8.45) with the O(a)/m terms neglected

8.2 Cause and Elimination of the Pre-Acceleration

93

d γ6 d γ (γu) − 2 (u) ˙ 2u (8.76a) dt dt c

dγ γ6 d dγ [Fext (t) + fa1 (t) + fa2 (t)]u 2 γ − 2 (u) (8.76b) = ˙ − τe mc2 dt dt dt c

d(γu) Fext (t)+fa1 (t)+fa2 (t) = − τe m dt

for γ(t)u(t) (and thus u(t) and u(t)) ˙ outside the transition regions where only the ﬁrst term (relativistic Newtonian acceleration) remains. Alternatively, the substitutions at the beginning of this chapter can be used to ﬁnd γ(t)u(t) from V(τ ) given in (8.75b). Either method yields for γu with Fext (t) = eE0 (as a → 0 with renormalized mass m) ⎧ 0, t<0 ⎪ ⎪ ⎨ d(γu) eE 0 1 , 0+ < t < t2 = γ 3 (t)u(t) ˙ = (8.77a) dt m ⎪ ⎪ ⎩ + 0 , t2 < t ⎧ 0 , t<0 ⎪ ⎪ ⎨ γ(t)u(t) = Δ(γu)1 + eE0 t/m , 0+ < t < t2 (8.77b) ⎪ ⎪ ⎩ Δ(γu)21 + eE0 t2 /m , t+ 2
+1/2 * γ(t) = 1 + [γ(t)u(t)/c]2

(8.77c)

and Δ(γu)21 = Δ(γu)2 + Δ(γu)1 , where Δ(γu)1 and Δ(γu)2 are the jumps in γ(t)u(t) across the transition intervals at t = 0 and t = t2 . (If desired, these jumps can be expressed in terms of ΔV1 and ΔV2 ; see (8.79) and (8.82) below.) One sees from this example of the motion of a charge through a parallelplate capacitor that the transition forces fan (t), which are nonzero only during the short time intervals following the points in time where the externally applied force is discontinuous, remove both the noncausal pre-acceleration and pre-deceleration from the solution to the equation of motion. However, the transition forces fan (t) in the equation of motion change, in general, the momentum and energy of the charged sphere. The next section determines conditions under which this change in momentum-energy is consistent with the conservation of momentum-energy and a non-negative radiated energy during the transition intervals. 8.2.5 Conservation of Momentum-Energy in the Causal Equation of Motion -N The transition forces fa (t) = n=1 fan (t) ensure that the solution to the modiﬁed equation of motion in (8.45) obeys causality while remaining free of runaway motion. However, these transition forces, in general, add to the momentum and energy of the charged particle. In particular, if the modiﬁed

94

8 Solutions to the Equation of Motion

power equation of motion (8.45b) with the O(a)/m terms neglected, or its rectilinear version in (8.76b), is integrated over time from before the external force is ﬁrst applied to a γΔta after the external force is turned oﬀ, the total work done by the external force should equal the total change in kinetic energy of the particle plus the total energy radiated by the charged particle. Since the total change in the reversible reactive Schott acceleration energy is zero and the change in kinetic energy is given by the change in γ, we have to assume that the remaining energy change across all the transition intervals (call it WTI ) equals the energy radiated during all the transition intervals, an energy that must be equal to or greater than zero. Before the external force turns oﬀ, the transition forces may contribute to reversible reactive acceleration energy as well as radiated energy and thus the remaining energy change across any one transition interval (call it WTI,n ) need not be equal to or greater than zero. Nonetheless, we shall ﬁrst determine the conditions under which this energy change WTI,n across a single transition interval is equal to or greater than zero. We see from (8.76b) that WTI,n across the nth transition interval from t = tn to t = tn + γΔta = t+ n as Δta → 0 is given by +

1 WTI,n = 2 mc mc2

tn

mτe γ 6 (u) ˙ 2 − fan u dt =τe [γ(t+ ˙ + ˙ n )] n )γ(t n ) − γ(tn )γ(t

tn

−[γ(t+ n ) − γ(tn )] .

(8.78)

The integral of the external force does not appear in the second equality of (8.78) because the work done by the ﬁnite external force during the transition interval approaches zero as Δta → 0. Since the acceleration of the extended charge can contain delta functions in the transition intervals as a → 0, the radiation from the extended charge during the transition intervals is no longer given by just the integration of mτe γ 6 (u) ˙ 2 but must include the integrations 7 of fan u as well. The right-hand side of the second equality in (8.78) contains 7

One may object to the presence of the fan u in the radiation integral of (8.78) on the grounds that Maxwell’s equations predict that the energy radiated by an accelerating point charge is given without this term [13], that is, by the mτe γ 6 (u) ˙ 2 term alone. However, as (8.32) shows, this derivation from Maxwell’s equations does not hold for an extended charge during the short time that it takes light to travel across the charge distribution, even as the radius of the charge approaches zero, because the velocity may become discontinuous as a → 0. (Abraham [3, sec. 25] and Hertz [44] were well aware that the energy radiated during a jump in velocity of an extended model of the electron was not given by the integral of the mτe γ 6 (u) ˙ 2 term in (8.78).) Even if the change in velocity across the transition interval is negligible, the textbook derivation and expression for the radiated energy from an accelerated point charge can become invalid for an extended charge distribution. To see this, consider a charge e with an acceleration given by δα (t), where the pulse width α of the ﬁnite-size delta function δα (t) is

8.2 Cause and Elimination of the Pre-Acceleration

95

the change in Schott acceleration energy minus the change in kinetic energy across the transition interval. Evaluating the right-hand side of (8.78) in terms of the proper time τ and velocity function V gives WTI,n τe = {V (τn+ )sinh[V(τn )/c + ΔVn /c] − V (τn ) sinh[V(τn )/c]} 2 mc c − cosh[V(τn )/c + ΔVn /c] + cosh[V(τn )/c] (8.79) where ΔVn is the jump in V across the transition interval. The values of V (τn ) and V (τn+ ) are determined solely by the externally applied force, whereas the jump ΔVn in the velocity function is determined solely by the transition force fan (τ ). The velocity function V(τn ) at the beginning of the transition interval can have any value from −∞ to +∞. Numerical evaluation of (8.79) reveals that for a value of the velocity function |V(τn )/c| 3 at the beginning of the transition interval, a ΔVn /c across the transition interval can be found to make WTI,n ≥ 0 for all values of V (τn+ ) if the value of acceleration at the beginning of the transition interval is restricted to |τe V (τn )/c| .9. As the value of |V(τn )/c| becomes smaller than 3, the value of |τe V (τn )/c| must become smaller than .9 to keep WTI,n ≥ 0 for all values of V (τn+ ). And if |V(τn )/c| 1, then |τe V (τn )/c| must be |V(τn )/c|/2; that is, if |V(τn )/c| 1 then |τe V (τn )/c| 1 to ﬁnd a ΔVn /c that makes WTI,n ≥ 0 for all values of V (τn+ ). Therefore, for N transition intervals between the time the external force -N is ﬁrst applied to when it is shut oﬀ, the total change in energy WTI = n=1 WTI,n , which equals the total energy radiated during the transition intervals and thus must be ≥ 0, can indeed be made ≥ 0 by choosing appropriate values for the jumps ΔVn across the transition intervals if the acceleration is restricted to values |V (τ )| τe 1, c

τ∈ / transition intervals .

(8.80)

It is easily proven, for example, from (8.61), that this inequality is satisﬁed if |Fext |τe 1. mc

(8.81)

greater than zero. The textbook expression for the total energy radiated by a point charge predicts a radiated energy proportional to e2 δα (t)dt = e2 . Now, distribute the charge e uniformly over the surface of a sphere of radius a. Each element de of this charge distribution will radiate a pulse with the pulse width α of the function δα (t). Therefore, for α a/c, the change in velocity across the transition interval is negligible, yet the ﬁelds radiated by diﬀerent elements of the extended charge distribution will hardly interfere, and the total radiated energy is approximately equal to the integrated sum of the energy radiated by 2 each element (which is proportional to (de) that the total energy radiated ), so by the extended charge is proportional to (de)2 → 0 = e2 . Moreover, the value of the total energy radiated by the extended charge of radius a can vary between 0 and the textbook value for a point charge as the pulse width α of δα (t) varies from α a/c to α a/c.

96

8 Solutions to the Equation of Motion

In the proper frame of the moving charged insulator, the inequality in (8.80) is identical to the condition in (8.24c) suﬃcient to neglect the O(a)/m terms in the proper-frame equation of motion because τe = 4a/(3c). Unfortunately, however, even if a is allowed to approach zero while renormalizing the mass m to a ﬁnite value (so that the conditions in (8.24) are all satisﬁed yet τe remains ﬁnite and equal to e2 /(6π0 mc3 )), the condition in (8.80) is still required to keep the energy radiated during the transition intervals equal to or greater than zero; see Section 8.5. Nonetheless, to within the approximation that the unrenormalized equation of motion in (8.76) and its solution are valid, WTI can always be made equal to or greater than zero by choosing the appropriate jumps in velocity across the transition intervals. Also, it can be shown that WTI is a small fraction of the total energy radiated if the amount of time that the external force is applied is several times greater than τe . Thus, there appears to be no inconsistency in assuming that WTI represents the energy radiated during all the transition intervals and that the unrenormalized causal equation of motion modiﬁed by the transition functions fan does not violate conservation of momentum-energy. For the example of the parallel-plate capacitor problem discussed in the previous section, we have for the total energy radiated across the two transition intervals + γΔt a =0

+

t2 +γΔt a =t2 1 mτe γ 6 (u) mτe γ 6 (u) ˙ 2 − fa1 u dt + ˙ 2 − fa2 u dt mc2 t2 0

ΔV1 eE0 τ2 ΔV1 − cosh = cosh + mc c c

eE0 τ2 ΔV2 ΔV1 − cosh + −1 + mc c c

eE0 τe eE0 τ2 ΔV1 ΔV1 − sinh − sinh (8.82) + mc mc c c

WTI 1 = mc2 mc2

where the values of V (τ ) and V(τ ) have been taken from (8.75). The ﬁrst line after the last equality sign in (8.82) gives the total work done on the charged particle by the external force. The second line after the last equality sign subtracts the total change in the kinetic energy of the charged particle. The third line after the last equality sign in (8.82) subtracts the energy radiated by the charged particle while it is outside the transition intervals. Using values of ΔV1 and ΔV2 given by ΔV1 /c = −ΔV2 /c = −C0 sign(eE0 ) ln[1 − |eE0 τe /(mc)|]

(8.83)

with the constant C0 > 1, the right-hand side of (8.82) is equal to or greater than zero for all values of the time τ2 that the constant external force eE0 is

8.2 Cause and Elimination of the Pre-Acceleration

applied to the charged sphere, provided |eE0 τe /(mc)| < 1; that is8 & & & eE0 τe & &<1. WTI ≥ 0 if && mc &

97

(8.84)

Moreover, (8.84) continues to hold if the initial value of the velocity is changed from zero to an arbitrary value. (If instead of the choice in (8.83), one chooses ΔV1 /c = −ΔV2 /c = eE0 τe /(mc), then WTI ≥ 0 if |eE0 τe /(mc)| 1, an inequality that conforms to the general criterion in (8.81) for ensuring that WTI ≥ 0.) Lastly, we determine the total momentum (GTI ) radiated by the charged particle across the two transition intervals by evaluating the diﬀerence between the total impulse imparted by the external force and the sum of the total change in kinetic momentum of the particle and the momentum it radiates outside the transition intervals. From (8.76a) one ﬁnds that for two transition -n=2 intervals, GTI = n=1 GTI,n , where +

tn 1 GTI,n = mτe γ 6 (u) ˙ 2 u/c2 − fan dt mc mc tn τe d d + + = γ(t+ ) )u(t )] − γ(t ) )u(t )] [γ(t [γ(t n n n n n n c dt dt 1 + − [γ(t+ n )u(tn ) − γ(tn )u(tn )] c which gives for the parallel-plate capacitor problem

eE0 τ2 ΔV1 ΔV1 GTI = sinh + − sinh mc mc c c

eE0 τ2 ΔV1 ΔV2 − sinh + + mc c c

eE0 τe eE0 τ2 ΔV1 ΔV1 − cosh − cosh . + mc mc c c 8

(8.85)

(8.86)

If |eE0 τe /(mc)| > 1 and ΔV1 has a ﬁnite value independent of τ2 , no values of ΔV1 and ΔV2 keep WTI ≥ 0 for all τ2 . If ΔV1 is allowed to depend on τ2 , then ΔV1 and ΔV2 can be chosen to keep WTI ≥ 0 for |eE0 τe /(mc)| > 1. This would violate causality, however, because the value ΔV1 for the jump in velocity when the external force is ﬁrst applied would have to anticipate the value of the time τ2 at which the force would be turned oﬀ. Another way to avoid the restriction in (8.84) on the magnitude of the external force is to allow the energy change WTI during a transition interval to become negative if the external force is large enough to not satisfy the second inequality in (8.84); for example, by postulating that the charged particle absorbs the energy that it radiated during a short time just prior to the transition interval. The physical mechanism for this re-absorption of radiated energy would remain unexplained, however, within the realm of classical physics.

98

8 Solutions to the Equation of Motion

The ﬁrst line after the equality sign in (8.86) is the total impulse imparted to the charged particle by the external force. The second line after the equality sign subtracts the total change in the kinetic momentum of the charged particle. The third line after the equality sign in (8.86) subtracts the momentum radiated by the charged particle while it is outside the transition intervals. For a physically realizable radiated momentum-energy, the absolute value of the radiated momentum multiplied by the speed of light c must be less than or equal to the radiated energy, that is & & & GTI & WTI & & (8.87) & mc & ≤ mc2 . Insertion of GTI /(mc) and WTI /(mc2 ) from (8.86) and (8.82), respectively, shows that the inequality in (8.87) is satisﬁed for all τ2 ≥ 0 with ΔV1 and ΔV2 given in (8.83) under the restriction on the value of the external force given in (8.84). Also, for eE0 τ2 /(mc) 1 the radiated momentum is greater than zero under the restriction on the external force in (8.84). If C0 ≥ 2 then the radiated momentum is greater than zero for all τ2 ≥ 0 under the restriction on the external force in (8.84). These results further conﬁrm the interpretation of WTI and GTI as energy and momentum, respectively, radiated by the charged sphere during the transition intervals. In the next section, the behavior of the solution to the equation of motion in the transition intervals is ignored and the method of successive substitutions is used to derive a power series solution to the rectilinear equation of motion and to the general equation of motion — the ﬁrst two terms of the latter power series equaling the Landau-Lifshitz approximation [51, sec. 76].

8.3 Power Series Solution to the Equation of Motion The pre-acceleration solution in (8.16) to the uncorrected rectilinear equation of motion (8.6) was derived in Section 8.1, and the solution (8.72) to the corrected rectilinear equation of motion (8.58) was derived in Section 8.2.2, under the assumption that the O(a)/m terms in (8.6) and (8.58) were negligible. (Of course, the O(a)/m terms vanish and τe remains a ﬁxed value if a → 0 while m is renormalized to a ﬁnite value.) In this section we shall use the method of successive substitutions to derive a power series solution to the uncorrected rectilinear equation of motion (8.6) and the uncorrected general equation of motion (7.1) or (7.12). (The uncorrected equations of motion omit the correction force fa (t) that exists only in the transition intervals following each nonanalytic point in time of the external force Fext (t) and that eliminates the noncausality from the exact solution to the uncorrected equations of motion; see Section 8.2. If the external force were an analytic function of complex time about the real time axis for all time from −∞ < t < +∞, no correction transition force would be required.) We begin with the derivation

8.3 Power Series Solution to the Equation of Motion

99

of a power series solution to the proper-frame equation of motion because of its relative simplicity. The equation of motion of the charged spherical insulator in its proper inertial frame of reference can be written from (5.12a) as u˙ =

Fext e2 ¨ + O(a2 ) + u mes 6π0 mes c3

(8.88)

wherein it is assumed that a is small enough that the mass mins of the insulator is negligible compared to the electrostatic mass mes = e2 /(8π0 ac2 ), which is ¨ and all the of order 1/a. Take the u˙ given in (8.88) and substitute it into u other derivatives of u on the right-hand side of (8.88). Repeat this substitution process successively to obtain the following power series solution for u˙ in the proper reference frame

n n ∞ e2 d Fext 1 + O(a3 ) . u˙ = 3 mes n=0 6π0 mes c dtn

(8.89)

In view of (8.24b)–(8.24d), the O(a3 ) terms in (8.89) are negligible if & & & n−1 & &d 2a && dn Fext && Fext && & (n + 1) & , n = 1, 2, . . . (8.90a) c & dtn & dtn−1 & and

a |Fext | c. c mes

(8.90b)

The conditions in (8.90a) state that the fractional changes in the externally applied force and its ﬁrst and higher order time derivatives are small during the time interval it takes light to traverse the radius of the charged sphere. Condition (8.90b) implies that the externally applied force is not large enough to change the velocity by a signiﬁcant fraction of c in the time interval light takes to traverse the charged sphere. If the externally applied force satisﬁes the conditions (8.90) so that the O(a3 ) terms are negligible, (8.89) can just as well be rewritten as

1 dFext e2 + O(a3 ) u˙ = Fext + (8.91) mes 6π0 mes c3 dt with the O(a3 ) terms in (8.91) also negligible under the conditions in (8.90). In general, dn Fext (t)/dtn will contain delta functions and their derivatives at t = 0 when the external force is ﬁrst applied and thus the power series solutions in (8.89) and throughout this section will not generally converge at t = 0. If the radius a of the charged sphere is allowed to approach zero in the equation of motion and the mass mes is renormalized to a ﬁnite value m, equations (8.88) and (8.89) reduce to

100

8 Solutions to the Equation of Motion

Fext ¨ + τe u m

(8.92)

∞ 1 n dn Fext τ m n=0 e dtn

(8.93)

u˙ = and u˙ =

with τe given in (8.7) as e2 /(6π0 mc3 ). Moreover, if the terms of higher order than n = 1 in (8.93) are negligible, that is & & &

& ∞ & & n & dFext & & n d Fext & & & (8.94) τe & & |Fext |, τe & & dtn & dt & n=2

then (8.93) can be approximated by

1 dFext u˙ ≈ Fext + τe . m dt

(8.95)

In other words, just the ﬁrst of the successive substitutions into (8.92) gives an accurate approximate solution to the proper-frame Lorentz-Abraham-Dirac equation of motion. For an electron, the conditions in (8.94) are satisﬁed except for changes in the external force that are so rapid that the eﬀect of these changes may not be accurately described by classical physics. 8.3.1 Power Series Solution to Rectilinear Equation of Motion The uncorrected equation for rectilinear motion in an arbitrary inertial reference frame for the charged insulator with m = mes can be expressed from (8.6) in the form V (τ ) =

Fext (τ ) e2 + V (τ ) + O(a2 ) . mes 6π0 mes c3

(8.96)

Applying the method of successive substitutions to (8.96), as we did to (8.88), we ﬁnd the power series solution for V (τ ), namely

n n ∞ e2 d Fext (τ ) 1 V (τ ) = + O(a3 ) mes n=0 6π0 mes c3 dτ n

which can be rewritten as

dFext (τ ) 1 e2 V (τ ) = + O(a3 ) . Fext (τ ) + mes 6π0 mes c3 dτ

(8.97)

(8.98)

The power series solution in (8.98) can be integrated with respect to the proper time to ﬁnd V(τ ) as

8.3 Power Series Solution to the Equation of Motion

⎡ τ ⎤

2 1 ⎣ e Fext (τ )⎦ + O(a3 ) . V(τ ) = Fext (τ )dτ + mes 6π0 mes c3

101

(8.99)

0

The solution in (8.98)–(8.99) for the rectilinear velocity function of the charged insulator of radius a contains no runaway solutions, no pre-acceleration, and is obtained using the single initial condition of zero velocity immediately before the external force is applied. Of course, an arbitrary constant velocity could be added to the right-hand side of (8.99) if the velocity were not zero before the external force were applied. However, regardless of the initial conditions the series solution in (8.97) contains spurious delta functions and their time derivatives at τ = 0 that violate the conditions (8.90) and do not satisfy the uncorrected equation of rectilinear motion (8.96) (or the corrected equation of motion (8.49) with the transition force given by (8.68)). The two terms in the brackets of (8.99) can also be found from the preacceleration solution (8.16) (with mes replacing m) by expanding the external force Fext (τ + τ ) in a Taylor series about the present time τ (assuming this expansion exists for τ > 0), so that integrating term by term yields [34], [52] ⎡ τ ⎤

n+1 n ∞ 2 1 ⎣ e d F (τ ) ext ⎦ . (8.100) V(τ ) = Fext (τ )dτ + 3 n mes 6π m c dτ 0 es n=0 0

However, this expansion (8.100) of the pre-acceleration integral in (8.16) does not, in general, yield a valid asymptotic series solution to (8.6) beyond the ﬁrst term in the summation of (8.100) because (8.16) was derived from the equation of motion (8.6) by neglecting self-force terms of order a2 . In other words, the O(a3 ) terms in (8.100) are not equal to the O(a3 ) terms in (8.99). (Recall that 1/mes = O(a).) It should also be noted that the power series solution (8.100) converges to the pre-acceleration solution (8.16) for τ > 0 but not for τ < 0. The reason for this discrepancy between the series solution (8.100) and the exact solution (8.16) to (8.6) with the O(a)/mes = O(a2 ) terms omitted is that Fext (τ + τ ) cannot be expanded in a Taylor series about τ ≤ 0 for all τ ≥ 0 because Fext (τ ) is identically zero for τ < 0. Also, to within a constant velocity, for τ > Δta the solutions to the corrected and uncorrected equations of motion are the same; compare (8.16) with (8.50) for τ > Δta . When the external force becomes zero after it is applied for a ﬁnite time interval, the power series solution (8.99), like the pre-acceleration solution (8.16), produces the same ﬁnal velocity that would be produced if the radiation reaction, the V term in (8.6), and all higher order terms were neglected. Also, like the pre-acceleration solution, the eﬀect of the radiation reaction on the power series solution for the velocity function V, during the time the external force is applied, approaches zero as aFext /mes as the radius a of the charged sphere approaches zero. Indeed, the motion of the charged insulator should be determined solely by the conventional momentum, mes d(γu)/dt, as the

102

8 Solutions to the Equation of Motion

radius of the shell approaches zero, since the mass mes becomes inﬁnite while the radiation reaction term remains ﬁnite as the radius a approaches zero. As long as Fext /mes remains nonzero, however, these results do not imply that the momentum and energy radiated after t = 0 e2 6π0 c5

t 6 2

γ u˙ u(t)dt

and

0+

e2 6π0 c3

t γ 6 u˙ 2 (t)dt

(8.101)

0+

respectively, for the power series solution of the charged insulator in rectilinear motion, approach zero as the radius a approaches zero. If the radius a of the charged sphere is allowed to approach zero and the mass mes is renormalized to a ﬁnite value m, (8.96) and (8.97) reduce to Fext (τ ) + τe V (τ ) m

(8.102)

∞ 1 n dn Fext (τ ) τ . m n=0 e dτ n

(8.103)

V (τ ) = and V (τ ) =

The terms of higher order than n = 1 in (8.103) can be neglected if & & &

& ∞ & & n & dFext (τ ) & & n d Fext (τ ) & & & τe & |Fext (τ )|, τe & & & & dτ n & dτ

(8.104)

n=2

and (8.103) can be approximated by

dFext (τ ) 1 V (τ ) ≈ Fext (τ ) + τe . m dτ

(8.105)

That is, just the ﬁrst of the successive substitutions into (8.102) gives an accurate approximate solution to the rectilinear Lorentz-Abraham-Dirac equation of motion. 8.3.2 Power Series Solution to General Equation of Motion The general equation of motion in an arbitrary inertial reference frame without the correction function fa (t) that eliminates the pre-acceleration can be written from (7.1) (with m replaced by mes ) as

2 i j d u e2 dui Fi i duj du + O(a2 ) . + u (8.106) = ext2 + ds mes c 6π0 mes c2 ds2 ds ds The method of successive substitutions applied to (8.106) produces the power series solution

8.3 Power Series Solution to the Equation of Motion

103

( . /) j i 1 ui Fext,j Fext dFext dui e2 i = + Fext + + O(a3 ) (8.107) ds mes c2 6π0 mes c2 ds mes c2 where Fext,j ≡ γ(−Fext , Fext · u/c) .

(8.108)

If the radius a of the charged sphere is allowed to approach zero while the mass mes is renormalized to a ﬁnite value m, the general equation of motion (8.106) reduces to the Lorentz-Abraham-Dirac (LAD) equation of motion

2 i i j Fext dui d u i duj du = + cτe +u ds mc2 ds2 ds ds

i Fext d2 ui d2 uj i = + cτe − u uj 2 mc2 ds2 ds i j Fext du (8.109) = + cτe Lij mc2 ds where Lij = (Iji − ui uj )(d/ds) and Iji is the four-vector identity matrix such that Iji Aj = Ai for any four vector Ai . The power series solution obtained by successive substitutions of dui /ds then becomes for the LAD equation of motion 1 i dui i j 2 i j k 3 i j k l = F + cτ L F + (cτ ) L L F + (cτ ) L L L F + · · · . e e e ext ext j j ext j l ext k k ds mc2 (8.110) If the third and higher order terms in (8.110) are negligible, that is, if & & & & &

& & β j & & & & β & β j k l + (cτe )3 Lβj Ljk Lkl Fext + · · · & &Fext & , cτe &Lj Fext & &(cτe )2 Lj Lk Fext (8.111) where the superscript β = (1, 2, 3) designates the three-vector part of a four vector, then (8.110) becomes dui 1 i j ≈ Fext + cτe Lij Fext 2 ds mc ( /) . j i dFext dFext 1 i i . Fext + cτe = − u uj mc2 ds ds

(8.112)

With the help of the relationship

duj j dFext,j dF j dF k 1 j uj ext = − Fext ≈ − 2 Fext,j + cτe − uj uk ext Fext ds ds mc ds ds dFext,j 1 j Fext = − 2 Fext,j + cτe (8.113) mc ds (8.112) can be re-expressed in the form

104

8 Solutions to the Equation of Motion

( . /) j i 1 ui Fext,j Fext dFext dui i ≈ + Fext + cτe ds mc2 ds mc2

(8.114)

& & & & dF β & & & β & & ext & cτe & & &Fext & & ds &

if

or, equivalently

(8.115a)

& & & d(γFext ) & & |Fext | . & τe & & dt

(8.115b)

The approximate solution in (8.114) is not always easy to apply in its present form because the external force on a charged particle is usually exerted by an external electromagnetic ﬁeld such that Fext (t) = e[E0 (r, t) + u(t) × B0 (r, t)]

(8.116)

the right-hand side of which involves u(t) and is a function not only of t but also of r, the position of the particle, so that dr/dt = u(t). Landau and Lifshitz [51, sec. 76] have partially alleviated this shortcoming of (8.114) by writing the external force in the four-dimensional form [13, eq. (18-32)]

F ij (r, t) =

Fij (r, t) =

i Fext = eF ij (r, t)uj

(8.117a)

Fext,i = eFij (r, t)uj

(8.117b)

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ cB0z

−cB0z

cB0y

0

−cB0x

⎪ −cB0y cB0x ⎪ ⎪ ⎪ ⎪ ⎩ −E −E0y 0x

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −cB0z

cB0z 0

0 −E0z

⎫ E0x ⎪ ⎪ ⎪ ⎪ ⎪ E0y ⎬

(8.118)

E0z ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎭

⎫ −cB0y −E0x ⎪ ⎪ ⎪ ⎪ ⎪ cB0x −E0y ⎬

⎪ cB0y −cB0x 0 −E0z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ −E −E −E 0 0x 0y 0z

.

Use of (8.117a) converts (8.114) to /) ( . j ij i dui u ) F F d(F 1 u j ext,j ext i . Fext + cτe e ≈ + ds mc2 ds mc2 e With d(F ij uj )/ds expressed as

(8.119)

(8.120)

8.3 Power Series Solution to the Equation of Motion

105

dF ij ∂F ij k d(F ij uj ) duj duj = uj + F ij = u uj + F ij ds ds ds ∂xk ds

(8.121a)

k dFext,j duj 1 uj Fext,k Fext ij F F + cτ ≈ + ext,j e ds mc2 ds mc2

(8.121b)

and F ij

which, in view of (8.115), can be approximated by F ij

duj τe i 1 ij k F F F + F F ≈ ext,j ext,k ext ext ds mc2 mce

(8.121c)

(8.120) becomes

1 τe2 dui ∂F ij k j i ≈ F 1 + + τ F F c e u uj ext,j e ext ext ds mc2 m2 c2 ∂xk

1 j . + (eF ij + ui Fext )Fext,j mc2 Under the additional assumption that & & 2 & & τe j & & & m2 c2 Fext,j Fext & 1 which is equivalent to

|Fext |τe 1 mc γ

(8.122)

(8.123a)

(8.123b)

(8.120) becomes equal to the approximate solution derived by Landau and Lifshitz [51, sec. 76]

dui ∂F ij k 1 1 i ij i j F + τe c e k u u j + . (8.124) (eF + u Fext )Fext,j ≈ ds mc2 ext ∂x mc2 This solution is an accurate approximation to the solution of the LAD equation of motion if the inequalities in (8.111), (8.115), and (8.123) are satisﬁed. Spohn [38] also concludes that the Landau-Lifshitz solution in (8.124) is eﬀectively the solution to the LAD equation of motion if the external force varies slowly on the scale of the parameter τe , that is, if the inequalities in (8.111), (8.115), and (8.123) are satisﬁed. However, it is emphasized that the Landau-Lifshitz solution in (8.124) is an approximate solution to the LAD equation of motion (8.109), as one can readily see by noting that the radiation reaction terms in (8.124) are not generally equal to the radiation reaction term in the LAD equation of motion (8.109) found by inserting dui /ds from (8.124) into the radiation reaction term of (8.109). Rohrlich [53] argues, nonetheless, that the Landau-Lifshitz solution in (8.124) is an accurate solution to the LAD equation of motion to within the limitations of classical physics imposed by quantum mechanics [51, p. 208], [54]–[55]; see last paragraph of 8.3.3.

106

8 Solutions to the Equation of Motion

The three-vector part of equation (8.124) can be expressed as [51, p. 213]

m d(γu) ∂ ∂ ≈ (E0 + u × B0 ) + τe γ + u · ∇ E0 + u × + u · ∇ B0 e dt ∂t ∂t eτe (u · E0 ) + E0 + c(E0 + u × B0 ) × B0 mc c eτe γ 2 (u · E0 )2 2 + u (8.125a) − |E + u × B | 0 0 mc2 c2 and the fourth component as

mc2 dγ ∂ ≈ u · E0 + τe γ u · + u · ∇ E0 e dt ∂t eτe 2 |E0 | + (u × B0 ) · E0 + m eτe γ 2 (u · E0 )2 2 − |E0 + u × B0 | + m c2

(8.125b)

after making use of the relations obtainable from (8.116)–(8.119), namely

u · E0 i (8.126a) Fext = eγ E0 + u × B0 , c

∂F ij k γ2 u · E0 u · ∇ E0 + u × B0 , u uj = ∂xk c c

∂E0 ∂B0 1 ∂E0 + +u× , u· (8.126b) ∂t ∂t c ∂t u · E0 ij F Fext,j = eγ E0 + c(E0 + u × B0 ) × B0 , (E0 + u × B0 ) · E0 c (8.126c) 2 (u · E ) 0 j Fext (8.126d) Fext,j = e2 γ 2 − |E0 + u × B0 |2 . c2 As a check, the power equation (8.125b) can also be obtained by taking the dot product of u with the force equation (8.125a). The approximate solution (8.125) to the LAD equation of motion can be derived quite easily from the three-vector equations of motion (7.12a) and (7.17a), which become as a → 0 and the mass is renormalized to a ﬁnite value m d(γu) d e2 γ4 2 ˙ m γ u˙ + 2 (u · u)u = Fext + dt 6π0 c3 dt c γ4 γ2 ˙ 2 + 2 (u · u) ˙ 2 u − 2 |u| (8.127a) c c

8.3 Power Series Solution to the Equation of Motion

107

d 3 u e2 ˙ . (γ u) (8.127b) + 2 c 6π0 c3 γ dt Substitute γ u˙ from (8.127b) into the right-hand side of (8.127a) to obtain the approximate solution

d(γu) d γ2 (u · Fext )2 2 m |F u . (8.128) | − ≈ Fext + τe (γFext ) − ext dt dt mc2 c2 mγ u˙ = Fext − (Fext · u)

The relationship dγ dFext dFext d (u · Fext )Fext +γ (γFext ) = Fext + γ ≈ dt dt dt mc2 dt converts (8.128) to d(γu) (u · Fext )Fext dFext m ≈ Fext + τe +γ dt mc2 dt

2 2 (u · Fext ) γ 2 u − 2 |Fext | − mc c2

(8.129)

(8.130a)

which possesses the corresponding power equation of motion 2 γ dFext 2 dγ 2 2 2 . (u · Fext ) − |Fext | u + γu · mc ≈ u · Fext + τe dt mc2 dt (8.130b) With Fext = e(E0 + u × B0 ) and

dFext dB0 dE0 +u× + u˙ × B0 (8.131) =e dt dt dt dE0 u dB0 1 ≈e E0 + u × B0 − (u · E0 ) 2 × B0 +u× + dt dt γ c (8.130a) becomes

m d(γu) dE0 dB0 ≈ (E0 + u × B0 ) + τe γ (8.132a) +u× e dt dt dt eτe (u · E0 ) + E0 + c(E0 + u × B0 ) × B0 mc c 2 eτe γ (u · E0 )2 2 + u. − |E + u × B | 0 0 mc2 c2

Taking the dot product of u with (8.132a) produces the power equation of motion mc2 dγ dE0 ≈ u · E0 + τe γu · (8.132b) e dt dt eτe |E0 |2 + (u × B0 ) · E0 + m eτe γ 2 (u · E0 )2 2 − |E0 + u × B0 | . + m c2

108

8 Solutions to the Equation of Motion

These two equations in (8.132) become identical to the ones in (8.125) when the total time derivatives of E0 and B0 are rewritten as

∂ d ∂ d E0 (r, t) = + u · ∇ E0 (r, t), B0 (r, t) = + u · ∇ B0 (r, t) . dt ∂t dt ∂t (8.133) If desired, the partial derivatives with respect to time in (8.133) and (8.125) can be written from Maxwell’s equations as ∂ E0 (r, t) = c2 ∇ × B0 (r, t) , ∂t

∂ B0 (r, t) = −∇ × E0 (r, t) . ∂t

(8.134)

For rectilinear motion the equations in (8.130) reduce to mγ 2 u˙ ≈

dFext Fext + τe γ dt

(8.135)

which agrees with (8.105). Note that for hyperbolic motion discussed in Section 7.1, the external force is constant and the Landau-Lifshitz solution in (8.135) is identical to the exact solution in (7.25)–(7.27). In the following section the Landau-Lifshitz approximation in (8.125) is solved explicitly for a charged particle moving in a uniform magnetic ﬁeld. 8.3.3 Charge Moving in a Uniform Magnetic Field The Landau-Lifshitz approximation (8.124) or (8.125) to the solution to the LAD equation of motion has been determined explicitly by Spohn [38] for a number of cases including that of a charge moving in a uniform magnetic ﬁeld z B0 = B0ˆ

(8.136)

where B0 is the constant magnitude of the magnetic induction. To simplify the derivation, assume that the velocity of the charge is zero in the z direction. With E0 = 0 and B0 given in (8.136), the approximate equations of motion in (8.125) reduce to m

e2 B02 τe γ 2 d(γu) z− ≈ eB0 u × ˆ u dt m

(8.137a)

and

dγ e2 B02 τe γ 2 2 (8.137b) ≈− u . dt m The power equation (8.137b) can also be found by dotting u into (8.137a). Next, cross u into (8.137a) to get mc2

mγu × u˙ ≈ −eB0 u2ˆ z. Taking the magnitude of both sides of this equation further gives

(8.138a)

8.3 Power Series Solution to the Equation of Motion

˙ ≈ eB0 u2 . mγ|u × u|

109

(8.138b)

Since the instantaneous radius of curvature of the trajectory of a particle moving in a plane is given by R(t) =

u3 ˙ |u × u|

(8.139)

(8.138b) combines with (8.139) to yield the instantaneous radius of curvature as mγu . (8.140) R(t) ≈ eB0 Writing u2 in terms of γ 2 in (8.137b) results in the ﬁrst order diﬀerential for γ e2 B02 τe 2 dγ ≈− γ −1 (8.141) dt m2 which has the solution γ−1 γ − 1 −2αt ≈ 0 e (8.142) γ+1 γ0 + 1 or γ + tanh(αt) (8.143) γ(t) ≈ 0 1 + γ0 tanh(αt) where

e2 B02 τe (8.144) m2 and γ0 = γ(0) is the initial value of γ(t) related to the initial speed u0 = u(0) of the particle by 1 γ0 = . (8.145) 1 − u20 /c2 α=

From the expression for γ(t) in (8.143) we ﬁnd that the speed u(t) of the charged particle is given by u(t) ≈

γ0 u0 sech(αt) . γ0 + tanh(αt)

(8.146)

Therefore, the instantaneous radius of curvature of the particle given in (8.140) becomes R0 sech(αt) R(t) ≈ (8.147) 1 + γ0 tanh(αt) with the initial radius of curvature given by R0 = mγ0 u0 /(eB0 ). The kinetic energy (W) of the particle is given by W(t) = mc2 γ(t) ≈ W0

γ0 + tanh(αt) γ0 [1 + γ0 tanh(αt)]

with the initial kinetic energy given by W0 = mc2 γ0 .

(8.148)

110

8 Solutions to the Equation of Motion

For t 1/α the speed and instantaneous radius of curvature decay exponentially as 2γ0 u0 −αt e u(t) ∼ (8.149) γ0 + 1 and R(t) ∼

m 2R0 −αt ∼ u(t) . e γ0 + 1 eB0

(8.150)

For t 1/α the speed, instantaneous radius of curvature, and kinetic energy of the particle are given approximately as

α t (8.151) u(t) ∼ u0 1 − γ0 R(t) ∼ R0 (1 − αγ0 t)

αγ0 u20 W(t) ∼ W0 1 − t . c2

(8.152) (8.153)

Note from (8.152) and (8.153) that the fractional change in radius of curvature and kinetic energy per unit time are approximately equal when the speed of the charge is approximately equal to the speed of light — in agreement with Shen’s results [56]. The expression (8.153) predicts a kinetic energy loss per revolution (with the period of revolution approximated by T0 = 2πR0 /u0 ) that agrees exactly with Plass’s result [52, eq. 147], and approximately with Schwinger’s result [57, eq. I.10] when the speed of the charge equals approximately the speed of light. The solution (8.124) for the speed, instantaneous radius of curvature, and kinetic energy given in (8.146)–(8.148) for a charge moving perpendicular to a uniform magnetic ﬁeld is still an approximate solution because (8.124) is an approximate solution to the Lorentz-Abraham-Dirac equation of motion (8.109). The approximation will be accurate if the inequalities in (8.111), (8.115), and (8.123) are satisﬁed. In particular, the inequality in (8.115b) becomes & & & d(γu) & e2 τe B02 u eτe B0 && (8.154a) ×ˆ z&& ≈ eB0 u dt m or, in terms of B0 B0

m eτe

(8.154b)

and the inequality in (8.123b) gives eτe B0 u 1 mc γ or m B0 eτe

3 1−

(8.155a)

u20 c2

(8.155b)

8.3 Power Series Solution to the Equation of Motion

111

since u/c < 1 and u ≤ u0 . This last inequality, which is more restrictive than the inequality in (8.154b), can be proven to be a necessary as well as suﬃcient condition for the Landau-Lifshitz solution to be a good approximation to the exact LAD solution by comparing the approximate dγ/dt in (8.141) with the exact dγ/dt given from (7.17b) as

2 dγ d γ γ3 2 ˙ . (8.156) − 2 |u| = τe γ dt dt2 c Note that Fext · u = eB0 (u × ˆ z) · u = 0 in (7.17b). If the approximate solution in (8.137) is accurate, it can be inserted into the right-hand side of (8.156) to get a quantity that is approximately equal to the right-hand side of (8.141). From the approximate solution in (8.137) we ﬁnd d2 γ dγ = 2α2 γ(γ 2 − 1) ≈ −2αγ 2 dt dt ˙ 2≈ |u|

αc2 (1 + ατe )(γ 2 − 1) τe γ 4

(8.157a) (8.157b)

which, when substituted into (8.156), yields dγ ≈ −α(γ 2 − 1)[1 − ατe (2γ 2 − 1)] . dt

(8.158)

Comparing (8.158) with (8.141), we see that the Landau-Lifshitz solution is a good approximation to the exact solution of the LAD equation of motion if and only if ατe γ 2 1 (8.159) which can be re-expressed as m B0 eτe

3 1−

u20 . c2

For an electron (8.160) becomes approximately 3 u2 12 1 − 20 Tesla. B0 10 c

(8.160)

(8.161)

This means that the solution given in (8.146)–(8.148) for the motion of a slowly moving electron (u20 /c2 1) in a uniform magnetic ﬁeld becomes an inaccurate approximation to the exact solution of the LAD equation of motion (8.109) if the value of the magnitude of the uniform magnetic ﬁeld is greater than about 1011 Tesla (1015 Gauss). This enormous value is on the order of the estimated magnetic ﬁelds in neutron stars called magnetars, which have the highest known magnetic ﬁelds in the universe. For extremely high energy

112

8 Solutions to the Equation of Motion

electrons, (8.161) implies that the approximate solution in (8.146)–(8.148) becomes inaccurate for much smaller values of the applied magnetic ﬁeld. Thus, the trajectory of extremely high energy charged particles in observable magnetic ﬁelds may reveal diﬀerences between the Landau-Lifshitz and exact solutions as well as limitations on the applicability of the LAD equation of motion. For the slowly moving charge in a uniform magnetic ﬁeld, the condition in (8.160) for the Landau-Lifshitz solution (8.124) to be an accurate solution to the LAD equation of motion (8.109) is equivalent to the condition in (8.81) (and (8.170) of Section 8.5 below) suﬃcient for the corrected LAD equation of motion (8.168) to satisfy causality while maintaining a non-negative radiated energy during the transition intervals when the external force is applied for a ﬁnite time. These comparable limiting conditions on the corrected LAD equation of motion and the Landau-Lifshitz approximate solution to the original (uncorrected) LAD equation of motion is compatible with Rohrlich’s contention [53] that the diﬀerence between the Landau-Lifshitz approximate solution and the exact solution to the LAD equation of motion lies within the limitations of classical physics imposed by quantum mechanics.

8.4 The Finite Diﬀerence Equation of Motion It was shown in Section 8.2 that if the evaluation of the self electromagnetic force is done properly near the time the external force is ﬁrst applied, a correction force fa (t) must be added to the original equation of motion during the short time interval 0 ≤ t ≤ Δt. With the proper choice of fa (t), this slight modiﬁcation removes the pre-acceleration from the solution to the uncorrected equation of motion. Power series solutions obtained in Section 8.3 to the original uncorrected equation of motion also eliminate the pre-acceleration, but at the expense of introducing spurious delta functions and their derivatives that do not satisfy either the uncorrected or corrected equation of motion near the time the external force is ﬁrst applied. Through the years a number of other methods have been proposed to eliminate the pre-acceleration that arises in the solution to the original uncorrected equation of motion (7.1) [58]–[62]. However, none of these alternative methods have been entirely successful because they either eliminate a priori all derivatives of acceleration [60]–[62], [63] or they sum inﬁnite series expansions that neglect nonlinear terms [58]–[59]. These latter methods [58]–[59] that have been proposed to eliminate the pre-acceleration or runaway solutions from the equation of motion involve determining explicitly the inﬁnite series of O(a) terms in the self electromagnetic force in (3.3) of the moving charged insulator of radius a. Speciﬁcally, Page [17] wrote down, without showing the derivation, this inﬁnite series and summed it in closed form to reveal that the self electromagnetic force in the proper frame of reference of the charge can be expressed as

8.4 The Finite Diﬀerence Equation of Motion

Fem (t) =

e2 u(t − 2a/c), u = 0 12π0 a2 c

113

(8.162a)

or, in an inertial frame in which the charge is moving with nonzero velocity much less than the speed of light, as Fem (t) =

e2 [u(t − 2a/c) − u(t)], (u/c)2 1 12π0 a2 c

(8.162b)

provided all nonlinear terms involving products of the time derivatives of the velocity are neglected and the correction force fa (t), explained in Section 8.2, is ignored. Equations (8.162) can also be found by discarding all but the ﬁrst series in the double inﬁnite series that Schott [64] derived for the self electromagnetic force on the noncontracting sphere (Abraham’s nonrelativistically rigid model rather than Lorentz’s relativistically rigid model of the electron). The inﬁnite number of discarded series involve nonlinear products in Schott’s expression that would change for the relativistically rigid model of the electron; however, the linear ﬁrst series is the same for both relativistically and nonrelativistically rigid models of the electron. ( A simple proof of (8.162) is given in Appendix D.) When the self electromagnetic force (8.162b) is used in the derivation of the equation of motion given in Chapter 5, we obtain e2 [u(t − 2a/c) − u(t)], (u/c)2 1 12π0 a2 c (8.163) for the nonrelativistic equation of motion. Again, the nonlinear product terms have been neglected in (8.163), and the negative bare mass M0 is given as −mes /3 in (5.11). (If the bare mass were omitted in (8.163), the rest mass of the charged shell would not equal mins + mes .) A relativistic generalization of the ﬁnite diﬀerence equation (8.162b) has been obtained by Caldirola [59], [65]. Similarly, a relativistic generalization of (8.163) to an arbitrary inertial i ˙ reference frame can be found by replacing Fext (t) with Fext (s), u(t) with i i 9 du (s)/ds, u(t − 2a/c) with u (s − 2a), and u(t) with g(s)ui (s), the scalar i ui (s) = 0. We then have factor g(s) inserted to make Fext ˙ Fext (t) = (mins + M0 )u(t) −

i Fext (s) = (mins + M0 ) c2

i dui (s) e2 u (s − 2a) − g(s)ui (s) (8.164a) − 2 ds 12π0 a

with 9

In the proper frame (u(t) = 0), the three-vector part of the relativistic general ization ui (s − 2a) reduces to u(t − 2a/c)/ 1 − u2 (t − 2a/c)/c2 rather than just u(t − 2a/c). However, to within the approximation that the nonlinear terms involving products of time derivatives of velocity are neglected in the proper frame, u(t − 2a/c)/ 1 − u2 (t − 2a/c)/c2 can be replaced by u(t − 2a/c).

114

8 Solutions to the Equation of Motion i Fext (s)ui (s) = 0 = ui (s)ui (s − 2a) − g(s) . i

(8.164b)

j

so that g(s) = ui (s)u (s − 2a) = uj (s − 2a)u (s) and i dui e2 u (s − 2a) − ui (s)uj (s)uj (s − 2a) . − 2 ds 12π0 a (8.165) Notwithstanding the appealing simplicity of the ﬁnite diﬀerence equation (8.163) and its relativistic generalization (8.165), there is little justiﬁcation to accept them as valid equations of motion that are accurate beyond the usual radiation reaction terms, since (8.162) and (8.163) neglect all nonlinear product terms (involving derivatives of velocity), which are not necessarily ¨ proper-frame negligible for the Lorentz model of the electron beyond the u radiation reaction term. It can be shown that the nonlinear and linear parts of the self electromagnetic force are both zero for certain radiationless motion of a nonrelativistically rigid spherical shell, namely, when the shell oscillates with an amplitude smaller than its radius and a period equal to 2a/c [66]–[68]. These radiationless oscillations with the self electromagnetic force (8.162) equal to zero would not, in general, be self sustaining, that is, Fext (t) would not equal zero in (8.163) except for the special case of mins +M0 equal to zero. (For Lorentz’s relativistically rigid model of the electron, Pearle [68] has shown that bounded radiationless motions do not exist.) The work of Herglotz [39] and Wildermuth [40], discussed in Section 8.2, would suggest that the ﬁnite diﬀerence (linearized) equation of motion (8.163) does not, in general, eliminate the pre-acceleration, that is, runaway solutions for t < 0. This can be demonstrated for rectilinear motion by letting the velocity in (8.163) have exp(qt) time dependence when Fext (t) is set equal to zero. The equation that results for q, when the material mass of the insulator is negligible, is then e−2aq/c = 1 − aq/(2c) (8.166) i Fext = (mins + M0 ) c2

which has the positive real solution q≈

c 2(1 − 5e−4 ) c ≈ 1.96 . 1 − 4e−4 a a

(8.167)

(If the mass of the insulator is not negligible, the equation for q also has a real positive solution provided a is small enough for the value of mins + M0 to be negative.) This failure of the ﬁnite diﬀerence equation of motion (8.163) to eliminate the homogeneous runaway solutions (so that pre-acceleration will still arise when the asymptotic condition in (8.12) is applied), coupled with the fact that the ﬁnite diﬀerence equation (8.163) neglects all nonlinear terms involving products of the time derivatives of velocity, leaves little reason to prefer (8.163) or its relativistic generalization (8.165) to the equation of motion that simply neglects the O(a) terms in (7.1). Moreover, like (7.1) the ﬁnite diﬀerence equation of motion (8.163) neglects the transition force fai in the

8.5 Renormalization of the Equation of Motion

115

corrected equation of motion (8.43). And, as Section 8.2 shows, it is this small but important correction to the conventional equation of motion that eliminates the noncausal pre-acceleration.

8.5 Renormalization of the Equation of Motion The four-vector form of the Lorentz-Abraham equation of motion for a charged insulating sphere of radius a, charge e, and mass m equal -N to imes +mins is given that remove the in (7.1) and in (8.43) with the transition forces fai = n=1 fan pre-acceleration (and pre-deceleration) from the original equation of motion. If the radius a of the charged sphere is allowed to approach zero, while keeping the mass m at a ﬁxed value (mass renormalization), the contribution of the O(a) terms goes to zero (except possibly at nonanalytic transition points in time of the externally applied force) and the original equation of motion in (7.1) becomes identical to the Lorentz-Abraham-Dirac (LAD) equation of motion [12]. The equation of motion in (8.43) then becomes equal to the LAD equation of motion modiﬁed by transition forces in the proper time intervals Δta → 0 following the points in times where the externally applied force is not locally expandable in a power series; for example, at the time t = 0 when the external force is ﬁrst applied. This modiﬁed LAD equation of motion can be written from (8.43) as

2 i N i j e2 d u i i 2 du i duj du − . (8.168) fan = mc +u Fext + ds 6π0 ds2 ds ds n=1 i The transition forces fan in (8.168), which approach delta functions and their derivatives as a → 0, remove the noncausal pre-acceleration and pre-deceleration from the original LAD equation of motion. Moreover, the analysis of rectilinear motion in Section 8.2.5 shows that the jumps in velocity10 associated i with the transition forces fan (t) can be chosen to conserve momentum-energy in the equation of motion while maintaining a non-negative radiated energy during the transition intervals, provided the proper-frame acceleration outside the transition intervals is bounded by

˙ e |u|τ 1, c

τe =

e2 6π0 mc3

(8.169)

the same condition in (8.24c) for neglecting the O(a) terms in the equation of motion of the extended charged sphere before the mass is renormalized (and thus τe = 4a/(3c)). This condition is satisﬁed if the external force obeys the inequality 10

A consequence of letting a → 0, and thus Δta → 0, while renormalizing the mass to a ﬁnite value is that changes in velocity across the transition intervals Δta become abrupt.

116

8 Solutions to the Equation of Motion

|Fext |τe 1. (8.170) mc There is some justiﬁcation, even in classical physics, for renormalizing the mass mes + mins to a ﬁnite value m as a → 0 and mes = e2 /(8π0 ac2 ) → ∞ to obtain the equation of motion of a point charge. It was mentioned in Chapter 5 that mins may be negative because it can include gravitational and other attractive formation energies [20], [21]. Thus, as a → 0 it is conceivable that mins → −∞ and that lima→∞ (mes + mins ) = m, the measured rest mass. It is especially disconcerting, therefore, that for the renormalized causal equation of motion (8.168), the restriction in (8.169) on the magnitude of the acceleration, or in (8.170) on the magnitude of the externally applied force, is needed to ensure this equation of motion satisﬁes conservation of momentum-energy while keeping the value of the energy radiated during the transition intervals equal to or greater than zero. For the extended charged sphere, the condition (8.169) is understandable because it is the same as the condition in (8.24c) needed, in general, to neglect the O(a) terms in the proper-frame equation of motion (and to ensure that the inﬁnite series comprising O(a) converges). This condition merely implies that the O(a) terms may not be negligible if the speed of the charged sphere changes by an appreciable fraction of the speed of light in the time it takes light to cross the sphere. As a → 0 and the mass is renormalized to a ﬁnite value m, however, the conditions in (8.24) are all satisﬁed except possibly at the nonanalytic transition points in time of the external force where the ﬁrst and higher order derivatives of the velocity can become singular. One would hope that accounting for these singularities at the i transition intervals as a → 0 by the transition forces fan in the modiﬁed LAD equation of motion (8.168) would rectify the equation of motion regardless of the magnitude of the externally applied force. This is not the case, however, if the external force is large enough to disobey (8.170) because then we have shown that the value of the energy radiated during the transition intervals can become negative, that is, the momentum-energy of the charged particle is not conserved by the equation of motion while maintaining a non-negative radiated energy during the transition intervals; see Footnote 8. For an electron in an external electric ﬁeld E, the inequality in (8.170) is satisﬁed unless E mc/(eτe ) = 6π0 m2 c4 /e3 = 2.7 × 1020 Volts/meter, an enormously high electric ﬁeld. Nonetheless, an equation of motion of a mass-renormalized point charge that is both causal and conserves momentumenergy while avoiding a negative radiated energy during the transition intervals no matter how large the value of the external force does not result by simply equating the sum of the point-charge radiation reaction force and the externally applied force to the relativistic Newtonian acceleration force (measured rest mass times relativistic acceleration) and inserting delta-function transition forces at the nonanalytic points in time of the external force to obtain (8.168). A causal classical equation of motion of a mass-renormalized point charge that also conserves momentum-energy with a non-negative radiated energy during the transition intervals for arbitrarily large values of the

8.5 Renormalization of the Equation of Motion

117

external force, if it exists, must involve a more complicated combining of the Newtonian and radiation reaction forces with the externally applied force than just a summation.11 It seems prudent, therefore, to simply accept (8.168) as the classical causal equation of motion of a mass-renormalized point charge under the restriction in (8.170) on the magnitude of the externally applied force, or to tolerate the noncausality in the original LAD equation of motion i given by (8.168) without the transition forces fan , especially since these transition forces lead to jumps in velocity across the transition intervals that are not uniquely determined by the externally applied force. Ultimately, a fully satisfactory equation of motion of a mass-renormalized point charge may require a uniﬁed theory of inertial and electromagnetic forces as well as the introduction of quantum eﬀects. Renormalization of the mass of the charged sphere as its radius shrinks to zero is an attempt to extract the equation of motion of the point electron from the classical self electromagnetic forces of an extended charge distribution. Such attempts, as Dirac wrote [69], “bring one up against the problem of the structure of the electron, which has not yet received any satisfactory solution.”

11

To recapitulate the essence of the argument, if the external force Fext (t) is an analytic function of complex t in a neighborhood of the real t axis except at the time t = 0 when the external force is ﬁrst applied to the charge e, we have proven from the Maxwell-Lorentz equations that as a → 0 the radiation reaction four-force equals [e2 /(6π0 )](d2 ui /ds2 +ui duj /ds duj /ds) for all t > 2γa/c = 0+ . Therefore, if the external four-force plus the radiation reaction four-force equals mc2 dui /ds, where m is the measured rest mass of the charge, then 2 i j d u dui e2 i i duj du Fext + + u (8.171) − mc2 = 0 during t > 0+ 2 6π0 ds ds ds ds for a point charge e with ﬁxed rest mass m. Between t = 0 and t = 0+ , the left-hand side of (8.171) is not equal to zero and its values during this short time interval that approaches zero as a → 0 cannot generally be determined by the Maxwell-Lorentz equations (because of the unknown time dependence of the velocity during this interval). Yet, if a solution exists, the left-hand side of (8.171) must equal some function (or generalized function such as a delta function) i between t = 0 and t = 0+ that can be denoted by −fa1 (s) and that is needed to preserve causality. If the external force has N nonanalytic points in time, i there are N transition distribution functions fan (s) needed to preserve causality and the resulting equation of motion is given in (8.168). Since we have proven that causal solutions to (8.168) can violate the requirement of momentum-energy conservation with non-negative radiated energy during the transition intervals if the magnitude of the external force is not restricted by the inequality in (8.170), it follows that (8.171) cannot hold exactly for extremely large external forces. In other words, the generalization of Newton’s second law of motion to classical point charges with renormalized mass is incompatible with the Maxwell-Lorentz equations and conservation of energy if the magnitude of the externally applied force becomes too large. The renormalized causal classical equation of motion of a point charge encounters a “high acceleration catastrophe.”

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62. E. Marx: Electromagnetic energy and momentum from a charged particle. International J. of Theoretical Physics, 14, pp 55–65 (1975) 63. J. Huschilt, W.E. Baylis: Solutions to the “new” equation of motion for classical charged particles. Phys. Rev. D, 9, pp 2479–2480 (1974) 64. G.A. Schott: The theory of the linear electric oscillator and its bearing on the electron theory. Phil. Mag., 3, pp 739–752 (1927) 65. F. Rohrlich: Classical self-force. Phys. Rev. D, 60, pp 084017-1–5 (1999) 66. G.A. Schott: The electromagnetic ﬁeld of a moving uniformly and rigidly electriﬁed sphere and its radiationless orbits. Phil. Mag., 15, 752-761 (1933); and The uniform circular motion with invariable normal spin of a rigidly and uniformly electriﬁed sphere, IV. Proc. Roy. Soc. Lond., A159, pp 570–591 (1937) 67. D. Bohm, M. Weinstein: The self-oscillations of a charged particle. Phys. Review, 74, pp 1789–1798 (1948) 68. P. Pearle: Absence of radiationless motions of relativistically rigid classical electron. Foundations of Physics, 7, pp 931–945 (1977) 69. P.A.M. Dirac: A new classical theory of electrons. Proc. Roy. Soc. Lond., A209, pp 291–296 (1951) 70. L. Page, N.I. Adams Jr.: Electrodynamics (D. Van Nostrand, New York 1940) 71. A. Pais: The early history of the theory of the electron: 1897-1947. In: Aspects of Quantum Theory, ed by A. Salam, E.P. Wigner (Cambridge University Press, Cambridge 1972) ch 5 72. S. Coleman: Classical electron theory from a modern standpoint. In: Electromagnetism: Paths to Research, ed by D. Teplitz (Plenum, NewYork 1982) ch 6 73. T.B. Hansen, A.D. Yaghjian: Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (IEEE/Wiley, New York 1999)

Appendices

A Derivation and Transformation of Small-Velocity Force and Power

In this appendix, we derive the proper-frame force equation of motion (3.3) and the small-velocity Lorentz power equation of motion (3.4) directly from the self electromagnetic force and power integrals of the spherical shell of charge. We then transform (3.3) relativistically to obtain the force equation of motion (2.1) for arbitrary velocity. A relativistic transformation of (3.4), however, leads to the erroneous result (3.5) for the power equation of motion rather than the power equation of motion (2.4). We also show that (2.4) does not transform covariantly, thereby conﬁrming that the general power equation of motion (2.4) is not produced by a relativistic transformation of the smallvelocity power equation of motion (3.4); see Section 3.1. Lorentz [4] and numerous modern physics texts, for example [13], [18], [34], have derived the Lorentz force equation of motion (3.3) in the proper (instantaneous rest) frame. But none, as far as I am aware, have directly derived the small-velocity power equation of motion (3.4), because it requires taking into account the variation of the velocity over the charge distribution. Of course, (3.4) could be obtained by letting u/c become much less than unity in the general power equation of motion (2.4), which was rigorously derived by Schott [16]. (As discussed in Chapter 3, Schott’s impressive derivation is so involved and lengthy that it discourages a detailed re-examination. Thus we provide an alternative, simpler, yet rigorous derivation of the general force and power equations of motion, (2.1) and (2.4), in Appendix B.)

A.1 Derivation of the Small-Velocity Force and Power A.1.1 Derivation of the Proper-Frame Force The self electromagnetic force on the spherical shell of charge in its proper (instantaneous rest) inertial frame of reference can be expressed by the Lorentz force integral in (3.1) with u(r, t) = 0, that is

122

Appendix A Small-Velocity Force and Power

Fem (t) =

E(r, t) de , u(r, t) = 0

(A.1)

charge

where the element of charge ρ(r, t)dV in (3.1) is relabeled de in (A.1). The self electric ﬁeld E(r, t) on the charge de at position r is produced by the remainder of the charge in the spherical shell. Speciﬁcally, the charge de at the position r (t ) produces an electric ﬁeld dE(r, t) given by [13] %

ˆ de u(r , t ) R ˆ ˙ × u(r , t ) dE(r, t) = × R − ! "3 R c2 c ˆ · u/c 4π0 1 − R ' 1 u2 (r , t ) , t ) u(r ˆ − (A.2) + 2 1 − R R c2 c ˙ , t ) refer to the velocity and acceleration of the charge where u(r , t ) and u(r de at the retarded time R (A.3) t = t − c that is dr (t ) u(r , t ) = (A.4) dt d2 r (t ) . (A.5) dt2 The vector R is deﬁned as the diﬀerence between the position r of de and the position r (t ) of de at the retarded time t ˙ , t ) = u(r

R = r − r (t ) .

(A.6)

˙ , t ) about the present time t, as When one expands R , u(r , t ), and u(r the radius of the charge shell becomes small, one obtains the following power series expansion of dE(r, t) in (A.2) % " 3 |u(r ˆ ˙ , t)|2 ˆ de 1 !ˆ R ˆ + u(r ˙ ˙ dE(r, t)= − , t) R , t) − R R · u(r 2 2 4π0 R 2c R 8 c4 ' 2 ˆ · u(r ˆ · u(r ˙ ˙ , t) ¨ [ R 3R , t)] , t) 3 2 u (r ˆ + ˙ , t) + + + O(R) (A.7) u(r R 4 c4 8 c4 3 c3 with R = r − r (t), and u(r , t) = 0. Equation (A.7) diﬀers from the corresponding expression in [13] where the dependence of R in (A.6) upon the retarded time is ignored. Also, (A.7) diﬀers from the corresponding equation in [18] and [34] as well as [13] by including the spatial dependence of the acceleration and its time derivative over the charge distribution. Both of these diﬀerences vanish, as we shall see below, when (A.7) is integrated over de to

A.1 Derivation of the Small-Velocity Force and Power

123

get E(r, t) and then E(r, t) is integrated over de in (A.1) to get the self electromagnetic force and the Lorentz force equation of motion. These diﬀerences do not vanish in the subsequent derivation of the self electromagnetic power and thus cannot be ignored in the derivation of the power equation of motion. ˙ , t) of the charge de at the position r (t) can be The acceleration u(r ˙ written in terms of the acceleration u(t) of the center of the shell by using the requirement of special relativity that the spherical shell contracts to an oblate spheroid (to order R2 ) as the speed increases. Speciﬁcally, we ﬁnd for u(r , t) = 0 ˙ r · u(t) ˙ , t) = u(t) ˙ ˙ u(r − (A.8) u(t) + O(R2 ) c2 and ¨ (r , t) = u ¨ (t) + O(R) . u (A.9) Substituting (A.8) and (A.9) into (A.7) and integrating over de gives the ﬁnal ˙ and the form for the self electric ﬁeld at (r, t) in terms of the acceleration (u) time derivative of acceleration (¨ u) of the center of the shell of charge1 ! % ˆ " r · u˙ 1 1 R ˆ · u) ˆ + u˙ ˙ E(r, t) = + − 1 ( R R 4π0 R2 2c2 R c2 charge

' " 3(R ˆ · u) ˆ ! ˙ ˙ 3R u 2¨ u 2 2 ˆ · u) ˙ − |u| ˙ + (R + + 3 + O(R) de , u = 0. (A.10) 8 c4 4c4 3c Next insert the self electric ﬁeld from (A.10) into (A.1) and perform the double integration over the shell of charge. All the terms with an odd number ˆ or r integrate to zero and the remaining even product terms of products of R integrate to give the familiar expression for the self electromagnetic force in the proper frame of reference Fem (t) = −

e2 e2 ˙ ¨ + O(a), u + u 6π0 ac2 6π0 c3

u=0.

(A.11)

Equating the sum of the externally applied force and the self electromagnetic force to zero, as Lorentz did in his original work [4], one obtains the Lorentz force equation of motion (3.3) in the proper frame of the spherical shell of charge. 1

Actually, the de integration should be over all charge except the charge de = ρdV on which we are calculating the self force. However, the ﬁeld produced by ρdV within its own volume dV approaches zero as dV → 0 for both stationary and moving charge [73, sec. 2.1.10]. Thus, the value of the integration over all charge is the same as its value with de omitted as de → 0.

124

Appendix A Small-Velocity Force and Power

A.1.2 Derivation of the Small-Velocity Power The power delivered to the moving charge by the self electromagnetic forces within the charge distribution is given by the charge integral in (3.2), namely u(r, t) · E(r, t)de (A.12) Pem (t) = charge

where again the element of charge ρ(r, t)dV in (3.2) is relabeled as de in (A.12). The velocity u(r, t) of the charge distribution in (A.12) is arbitrary. For small velocity, u(r, t) can be written in terms of the velocity and acceleration of the center of the shell by using the information that the spherical shell contracts to an oblate spheroid (to order R2 ) as the speed of the charge increases; speciﬁcally

2 r · u(t) u 2 ˙ (A.13) u(r, t) = u(t) − ,R . u(t) +O c2 c2 Repeating the derivation that led to (A.10), with small-velocity instead of zero velocity, shows that (A.10) also remains valid to order u2 /c2 , that is ! % ˆ " r · u˙ 1 1 R ˆ · u) ˆ + u˙ ˙ + − 1 ( R R E(r, t) = 4π0 R2 2c2 R c2 charge

+

ˆ ! 3R 8 c4

2 ' " 3(R ˆ · u) ˙ u˙ 2¨ u u 2 2 ˆ ˙ − |u| ˙ (R · u) + + 3 +O ,R de . 4 4c 3c c2

(A.14)

Substitution of E(r, t) from (A.14) and u(r, t) from (A.13) into (A.12) allows Pem (t) to be written as Pem (t) = u(t)·

2

ˆ u R [r · u(t)] ˙ E(r, t)de− u(t)· de de+O , a . (A.15) R2 4π0 c2 c2

charge

charge

The integral of the electric ﬁeld in (A.15) is just the self electromagnetic force given in (A.11). The second integral in (A.15) is the extra term that arises because the velocity of the charge distribution varies with position around the shell. It evaluates to ˆ e2 R(r · u(t)) ˙ −u(t) · de de = − u · u˙ . (A.16) 2 2 4π0 c R 24π0 ac2 charge

The self electromagnetic power can thus be written as e2 Pem (t) = u · Fem (t) − u · u˙ + O 24π0 ac2

u2 ,a c2

(A.17)

A.2. Relativistic Transformation of Small-Velocity Force and Power

125

or Pem (t) = −

5e2 e2 ¨ + O(a), u · u˙ + u·u 2 24π0 ac 6π0 c3

u2 1. c2

(A.18)

Setting the sum of the power delivered by the externally applied force Fext · u and the self electromagnetic power Pem (t) equal to zero, as Lorentz did in his original work [4], one obtains the power equation of motion (3.4) for charge shells with small velocity (u2 /c2 1).

A.2 Relativistic Transformation of the Small-Velocity Force and Power As explained in Section 3.1 the point relativistic transformations do not necessarily apply to the integrated force and power that comprise the right-hand sides of the Lorentz force and power equations of motion, (3.3) and (3.4), respectively. Thus, it is not mathematically rigorous to transform the smallvelocity equations of motion, (3.3) and (3.4), to obtain the corresponding equations of motion, (2.1) and (2.4), for an arbitrary center velocity of the charge distribution. Nevertheless, a relativistic transformation of the properframe force equation of motion (3.3) does yield the general force equation of motion (2.1); whereas, a relativistic transformation of the small-velocity power equation of motion (3.4) does not yield the general power equation of motion (2.4). The proofs of these results follow. A.2.1 Relativistic Transformation of the Proper-Frame Force Let K be the proper inertial reference frame in which equation (3.3) is derived, and K be the arbitrary inertial frame in which the velocity of the center of the charged shell is u . Thus K has velocity u with respect to K . Equation (3.3) can be divided into components parallel and perpendicular to the velocity u ¨ u˙ u e2 + O(a) (A.19a) Fext = − c 6π0 c2 a ¨⊥ u u˙ ⊥ e2 F⊥ − + O(a) . (A.19b) = ext 6π0 c2 a c From the relativistic transformation of force ¨ u˙ u e2 + O(a) Fext = Fext = − 6π0 c2 a c ¨⊥ u˙ ⊥ e2 u ⊥ Fext + O(a) = F⊥ /γ = − ext 6π0 c2 γ a c

(A.20a)

(A.20b)

126

Appendix A Small-Velocity Force and Power

−1/2 with γ = 1 − u2 /c2 . The relativistic transformation of acceleration and its time derivative u˙ = γ 3 u˙ (A.21a) u˙ ⊥ = γ 2 u˙ ⊥ ¨ = γ 4 u ¨ + u

3γ 6 2 |u˙ | u c2

(A.21b) (A.22a)

3γ (u · u˙ )u˙ ⊥ (A.22b) c2 substituted into (A.20) produce the equations in the arbitrary K system ( 3 ) 4 6 ¨ γ u˙ γ u e2 3γ 2 (A.23a) Fext = − − 3 |u˙ | u + O(a) 6π0 c2 a c c ¨⊥ = γ u ¨ ⊥ + u 3

⊥ Fext

e2 = 6π0 c2

(

5

) 2 4 γ γ u˙ ⊥ 3γ ¨ − 3 (u · u˙ )u˙ ⊥ + O(a) . − u a c ⊥ c

(A.23b)

Adding (A.23a) to (A.23b), combining terms and removing the primes, results in the transformed equation of motion 3γ 2 d e2 e2 γ 2 ¨ + 2 (u · u) ˙ u˙ Fext = u (γu) − 2 3 6π0 ac dt 6π0 c c γ2 3γ 2 ¨ + 2 (u · u) ˙ 2 u + O(a) + 2 u·u (A.24) c c which is identical to the general equation of motion (2.1) obtained from the self electromagnetic force calculated directly in an inertial frame in which the charge has arbitrary center velocity u. A.2.2 Relativistic Transformation of the Small-Velocity Power In an inertial frame K in which the charge has an inﬁnitesimally small center velocity u, we have from equation (3.4) ¨ 5u˙ e2 u Fext · u = · u + O(a), u → 0 . (A.25) − 6π0 c2 4a c In the K frame, moving with velocity −u with respect to K (as u approaches zero), the velocity of the particle is u . Thus, in the K frame (A.25) becomes ¨ u 5u˙ e2 − · u + O(a) . Fext · u = (A.26) 6π0 c2 4a c

A.3 Noncovariance of the Power Equation

127

˙ and u ¨ in (A.20a), (A.21a) From the relativistic transformations of Fext , u, and (A.22a), we ﬁnd Fext · u = Fext · u (A.27) and

) ( 3 ˙ 4 ¨ 6 ¨ 5u˙ 5γ u u u γ 3γ · u = − − − 3 |u˙ |2 u · u . 4a c 4a c c

(A.28)

Substituting (A.27) and (A.28) into (A.26) and removing the primes, we obtain the general power equation of motion (3.5) 3γ 2 5e2 dγ e2 γ 4 2 ¨ ˙ Fext · u = u · u + + O(a) (A.29) (u · u) − 24π0 a dt 6π0 c3 c2 corresponding to the small-velocity power equation of motion (3.4). Unlike the case of the transformed force equation of motion, (A.29) is not identical to the power equation of motion (2.4) obtained from the self electromagnetic power calculated directly in an inertial frame in which the charge has arbitrary center velocity u. As explained in Section 3.1, we cannot rigorously apply the point relativistic transformations to the small velocity self electromagnetic force and power expressions to ﬁnd the self electromagnetic force and power of an arbitrarily moving charge, because the charge is distributed over an extended region of space and not concentrated at a single point moving with a uniform velocity. The distributed charge motion does not change the ﬁnal result of the self electromagnetic force calculation, but does change the 1/a term in the self electromagnetic power calculation, and the transformation properties of the self electromagnetic power. Indeed, the next section of this Appendix A demonstrates that the power equation of motion (2.4) does not transform covariantly.

A.3 Noncovariance of the Power Equation Begin with the power equation of motion (2.4) in an arbitrary inertial frame Ka

1 e2 γ 4 3γ 2 e2 d 2 ¨ ˙ γ− − u · u + + O(a). (A.30) Fext · u = (u · u) 6π0 a dt 4γ 6π0 c3 c2 In an inertial frame Kw moving with velocity w with respect to Ka , the rela˙ u ¨ , and γ in terms of the corresponding tivistic transformations of Fext , u, u, frame recast (A.30) in the form primed variables in the Kw

1 d(γ u ) e2 0 = (u + w) · Fext − 1 − 2 2 2 2 2 6π0 ac dt 4γ γw (1 + u · w/c )

2 2 2 e2 γ 2 3γ γ 3γ 2 ¨ + 2 (u · u˙ )u˙ + 2 u · u ¨ + 2 (u · u˙ ) u + O(a) . u + 6π0 c3 c c c (A.31)

128

Appendix A Small-Velocity Force and Power

If (A.31) is to be independent of w and hold for all w (w < c), then the terms in the curly brackets of (A.31) must be zero, that is 1 d(γ u ) e2 Fext = 1 − (A.32) 2 (1 + u · w/c2 )2 6π0 ac2 dt 4γ 2 γw

2 2 2 e2 γ 2 3γ γ 3γ 2 ¨ + 2 (u · u˙ )u˙ + 2 u · u ¨ + 2 (u · u˙ ) u + O(a). − u 6π0 c3 c c c Because of the 1/4 term in (A.32), the form of this equation (A.32) depends explicitly on the velocity w of the Kw inertial frame. Thus the form of (A.32) is not relativistically invariant with respect to a change of inertial frames, that is, the left- and right-hand sides of the power equation of motion (2.4) do not transform covariantly because of the −[e2 /(24π0 a)]d(1/γ)/dt term. Of course, it is this very term that the internal binding forces eliminate from the power equation of motion (2.4); see Chapter 4.

B Derivation of Force and Power at Arbitrary Velocity

In this appendix the self electromagnetic force and power are derived from equations (3.1) and (3.2) for the shell of charge moving with arbitrary velocity. The 1/a terms are derived from the space integrals in (3.1) and (3.2) evaluated for arbitrary, time-varying velocity (unlike the traditional heuristic derivation which assumes a constant velocity charge). The radiation reaction terms are found from the charge (rather than the space) integrals in (3.1) and (3.2) evaluated for a shell of charge moving with arbitrary, time-varying velocity.

B.1 The 1/a Terms of Self Electromagnetic Force and Power The self electromagnetic force and power of the moving shell of charge can be written as space integrals of the electromagnetic ﬁelds of the moving charge [15, sec. 2.5, eq. (25) and sec. 2.19, eq. (6)] d ¯ · n ˆ dS Fem (t ) = −0 E (r , t ) × B (r , t )dV + T (B.1) dt (t ) = − Pem

0 d 2 dt

V

V

(E 2 + c2 B 2 )dV − 0 c2

S

ˆ dS (E × B ) · n

(B.2)

S

¯ is Maxwell’s stress tensor (dyadic) and the primes denote quantities where T in a K inertial frame in which the charge shell has arbitrary center velocity u (t ). The volume V is enclosed by the surface S, which encloses the moving charge distribution. The force on any part of the charged oblate spheroid (with major axis 2a and minor axis 2a(1 − u2 /c2 )1/2 in the K frame will be caused by the position of the rest of the charge at an earlier time. In particular, the force ﬁeld on the leading end of the particle will have left the trailing end of the particle in a time Δt given approximately for small radius a by

130

Appendix B Force and Power at Arbitrary Velocity

(c − u )Δt = 2a 1 − u2 /c2

4 or

2a Δt = c

1 + u /c . 1 − u /c

(B.3)

In this time interval Δt the charge will have traveled a distance Δd given approximately by 4 a 1 + u /c 2u Δd = u Δt = . (B.4) c 1 − u /c Equation (B.4) says that the motion of the charge, when the charge is farther away from its present position than some ﬁnite number times the radius a, will not aﬀect the self electromagnetic force calculation. Thus, we can assume, with no loss of generality in the derivation, that the charge had uniform velocity when evaluating the ﬁelds for r greater than La where L is an indeﬁnitely large but ﬁnite number. In other words, if we choose the radius of the surface S larger by a factor L then the major radius of the oblate spheroidal charge ¯ and the Poynting vector E ×B in the surface distribution, the stress tensor T integrals of (B.1) and (B.2) can be assumed those of a charge distribution moving with constant velocity. Because each of these surface integrals is zero for a constant velocity charge distribution, (B.1) and (B.2) can be written in terms of the volume integrals alone d Fem (t ) = −0 (B.5) E (r , t ) × B (r , t ) dV dt Va

(t ) = − Pem

0 d 2 dt

(E 2 + c2 B 2 ) dV

(B.6)

Va

with Va denoting a ﬁnite volume that encloses the charge distribution and having a radius La proportional to the dimension a of the charged shell. The fact that the radius La of the volume Va approaches zero as a approaches zero, and yet L is an indeﬁnitely large number, is used in the following evaluations of the 1/a terms of self force and power. B.1.1 Evaluation of 1/a Term of Self Electromagnetic Force We want to evaluate the space integral in (B.5) at each instant of time t . To begin, let this instant of time be t = 0, in order to simplify the integral in (B.5) to E (r , 0) × B (r , 0) dV .

IF =

(B.7)

Va

Next write the ﬁelds, E (r , 0) and B (r , 0) in the K frame in terms of the ﬁelds in a proper inertial frame K at rest instantaneously with the center of the charge distribution at t = 0. Assume that the origins of the K and K

B.1

The 1/a Terms of Self Electromagnetic Force and Power

131

frames coincide at t = t = 0. Then the relativistic transformations of the ﬁelds are given by

¯ · [E(r, t) − u × B(r, t)] E (r , 0) = α

(B.8a)

¯ · [B(r, t) + u × E(r, t)/c2 ] B (r , 0) = α

(B.8b)

¯

¯ = γ I + (1 − γ )ˆ ˆ, α uu with

2

2 −1/2

γ = (1 − u /c )

(B.8c)

r⊥ = r⊥

(B.9a)

γ r

(B.9b)

r =

t = −γ u · r /c2

(B.9c)

where the subscripts ⊥ and mean perpendicular and parallel to the center velocity u . Substitute (B.8) and (B.9) into (B.7) and make the change of integration variable r = r⊥ + γ r (B.10a) so that

dV = γ dV

(B.10b)

and (B.7) becomes 1 ¯ · E(r, t = −u · r/c2 ) − u × B(r, t = −u · r/c2 ) IF = α γ Va ¯ · B + u × E/c2 dV . ×α (B.11) Since we have determined in Appendix C the proper-frame electric and magnetic ﬁelds, E(r, t) and B(r, t), at a ﬁxed time t, the integral of the ﬁelds in (B.11) could be evaluated if it weren’t for the fact that t = −u · r/c2 is not ﬁxed but varies with the integration variable r. Fortunately, this diﬃculty can be overcome, when evaluating the 1/a term, by expanding E(r, t = −u · r/c2 ) and B(r, t = −u · r/c2 ) about the ﬁxed time t = 0; speciﬁcally E(r, t) = E(r, 0) +

∂E(r, 0) t + ··· ∂t

(B.12a)

∂B(r, 0) t + ··· . (B.12b) ∂t From Maxwell’s equations all the time derivatives of E(r, 0) and B(r, 0) can be written in terms of the spatial derivatives B(r, t) = B(r, 0) +

∂B(r, 0) = −∇ × E(r, 0) ∂t

(B.13a)

132

Appendix B Force and Power at Arbitrary Velocity

∂E(r, 0) = c2 ∇ × B(r, 0) (B.13b) ∂t ∂E(r, 0) ∂ 2 B(r, 0) = −c2 ∇ × ∇ × B(r, 0) = −∇ × (B.13c) ∂t2 ∂t and so on. Substitution of the time derivatives from (B.13) converts (B.12) to E(r, t) = E(r, 0) + c2 ∇ × B(r, 0)t + · · ·

(B.14a)

B(r, t) = B(r, 0) − ∇ × E(r, 0)t + · · · .

(B.14b)

When the proper-frame electric and magnetic ﬁelds, E(r, 0), B(r, 0) and their curls, are inserted from (C.1) and (C.5) of Appendix C into the righthand sides of (B.14), and the resulting ﬁelds, E(r, t) and B(r, t), are inserted into the integrand of (B.11) with t = −u · r/c2 , one ﬁnds that the B ﬁeld in the integrand of (B.11) does not contribute to the 1/a term of the integral and that only the static part of the E ﬁeld contributes to the 1/a term. In more detail 1 ¯ · E(r, 0)] × α ¯ · (u × E(r, 0))/c2 dV + O(1) IF = (B.15) [α γ Va (a→0)

¯

¯ · E = γ E + (1 − γ )(ˆ u · E)ˆ u or since α · (u × E) = γ u × E and α * 2 + IF = u γ E + (1 − γ )(ˆ u · E)2 − (u · E)E dV + O(1). (B.16) Va (a→0)

The electric ﬁeld E(r, 0) is found by integrating expression (C.1) to get ⎧ e ⎪ ˆr + O(1/r), r > a ⎨ 4π0 r2 (B.17) E(r, 0) = ⎪ ⎩ O(1/a), r
d e2 (γ u ) + O(1) 2 6π0 ac dt

(B.19)

B.2

Radiation Reaction of Self Electromagnetic Force and Power

133

B.1.2 Evaluation of 1/a Term of Self Electromagnetic Power Proceeding with the evaluation of the self power integral in (B.6) IP = (E 2 + c2 B 2 )dV

(B.20)

Va

in the same manner as in the previous section for the self force integral, one gets 1 ¯ · E(r, 0)|2 + γ |u × E |2 /c2 dV + O(1) . |α (B.21) IP = γ Va (a→0)

With E(r, 0) inserted from (B.17), (B.21) integrates for large L to

2u2 e2 1 1 γ + IP = − γ + 2 γ + O(1) 4π20 a 3 γ 2c

2 2 u e γ 1 + 2 + O(1) = 2 4π0 a 3c which, when inserted into (B.6), gives

u2 e2 d Pem (t ) = − γ 1+ 2 + O(1) 8π0 a dt 3c

(B.22)

(B.23a)

or equivalently (t ) Pem

e2 d =− 6π0 a dt

1 γ − 4γ

+ O(1)

(B.23b)

for the 1/a term of the self electromagnetic power in the arbitrary K frame.

B.2 Radiation Reaction of Self Electromagnetic Force and Power The above derivation for the 1/a terms of the self electromagnetic force and power in an arbitrary inertial frame from the momentum and energy integrals in (B.5) and (B.6) does not extend easily to ﬁnding the radiation reaction (O(1)) terms of the self force and power because an inﬁnite number of terms in the series expansion (B.14) of E(r, t) and B(r, t) contribute to the O(1) terms of the momentum and energy integrals. Fortunately, we can ﬁnd the radiation reaction terms of the self force and power from the charge integrals of (3.1) and (3.2).

134

Appendix B Force and Power at Arbitrary Velocity

B.2.1 Evaluation of the Radiation Reaction Force To determine the O(1) terms of the self electromagnetic force in a K inertial frame in which the shell of charge is moving with arbitrary velocity, we shall evaluate the charge integral in (3.1) at an arbitrary instant of time t . To reduce the algebra, let this arbitrary time be chosen as t = 0, initially, so the self electromagnetic force in the K frame can be written ρ (r , 0)[E (r , 0) + u (r , 0) × B (r , 0)] dV . (B.24) Fem = charge

The electric and magnetic ﬁelds, E (r , 0) and B (r , 0), in (B.24) can be expressed by means of the relativistic transformations (B.8) and (B.9), in terms of the ﬁelds in the proper frame K at rest instantaneously with the center of the charge distribution at t = 0. Since (C.7) of Appendix C can be used to show that B(r, t) in (B.8) contributes only to terms of higher order than O(1), E and B in (B.24) can be written simply from (B.8) as ¯ · E(r, t) E (r , 0) = α

(B.25a)

B (r , 0) = γ u × E(r, t)/c2 ¯

(B.25b)

where α is deﬁned in (B.8c), r and t are given in (B.9), and u is the velocity of the center of the charge distribution. The velocity u (r , 0) of the charge distribution in the K frame can be written in terms of the velocity u(r, t) in the proper frame by means of the relativistic transformation u(r, t)/γ + u u(r, t) · u (1 − 1/γ )/u2 + 1 u (r , 0) = . (B.26) 1 + u(r, t) · u /c2 Similarly, the charge density ρ (r , 0) in (B.24) transforms relativistically to the proper K frame as ρ (r , 0) = γ ρ(r, t) 1 + u(r, t) · u /c2 (B.27) with r and t again given in (B.9). The velocity u(r, t) and the charge density ρ(r, t) of the charge distribution at t = −γ u · r /c2 = −u · r/c2 in the K frame can be expanded about t = 0 to give ˙ 0)(u · r)/c2 + O(r2 ) u(r, t) = −u(r,

(B.28)

∂ρ(r, 0) u · r + O(r2 ) . (B.29) ∂t c2 Because u(r, 0) equals zero for a relativistically rigid, nonrotating charge distribution, ∂ρ(r, 0)/∂t = −∇ · [ρ(r, 0)u(r, 0)] = 0, and (B.29) becomes simply ρ(r, t) = ρ(r, 0) −

ρ(r, t) = ρ(r, 0) + O(r2 ) .

(B.30)

B.2

Radiation Reaction of Self Electromagnetic Force and Power

135

Substituting u(r, t) from (B.28) and ρ(r, t) from (B.30) into (B.27) gives ˙ 0)(u · r) u · u(r, ρ (r , 0) = γ ρ(r, 0) 1 − + O(r2 ) (B.31) c4 for the charge density in the K frame. Similarly, substituting (B.28) into (B.26) and expanding in powers of r gives ˙ 0)(u · r) u u u(r, ¯ u (r , 0) = u − Iγ + 2 (1 − γ ) · + O(r2 ) . (B.32) u γ 2 c2 ˙ 0) of the charge distribution in the proper frame was The acceleration u(r, ˙ thus (A.8) given previously in (A.8) in terms of its center acceleration u; shows that (B.31) and (B.32) remain valid to O(r2 ) when the acceleration ˙ 0) is replaced by the center acceleration u. ˙ u(r, Substitute into (B.24) the expressions (B.25) for E (r , 0) and B (r , 0), (B.31) for ρ (r , 0), (B.32) for u (r , 0) (all with r and t replaced from (B.9) ˙ 0) in (B.31) and (B.32)); then and the center acceleration u˙ replacing u(r, make the change of integration variable from r to r⊥ + r /γ to obtain ˙ · r) (u · u)(u γ 2 ¯ · E(r, t) + 2 u ρ(r, 0) 1 − + O(r ) α Fem (0) = c4 c charge

˙ · r) u u u(u 2 ¯ − Iγ + 2 (1 − γ ) · + O(r ) × (u × E(r, t)) dV u γ 2 c2

(B.33)

with t = −u · r/c2 . We want to insert E(r, t) from (B.14a) into the integrand of (B.33); speciﬁcally E(r, t) = E(r, 0) + c2 ∇ × B(r, 0)t − c2 ∇ × ∇ × E(r, 0)

t2 + ··· 2

(B.34)

with t = −u · r/c2 . When one replaces E(r, 0) and B(r, 0) in (B.34) by their integral values given in (C.1) and (C.5), one ﬁnds 1 E(r, t) = E(r, 0) + terms odd in ˆr + (terms even in ˆr) . a

(B.35)

As the radius a of the charged sphere approaches zero, the terms odd in ˆr in (B.35) integrate to zero in (B.33). The 1/a terms in (B.35) integrate to give 1/a terms when multiplied by the terms of order unity in the integrand of (B.33), and zero when multiplied by the terms of order r and higher in (B.33). Also, as a approaches zero, the O(r2 ) terms in (B.33) integrate to zero. In all, (B.33) becomes

˙ · r) 1 (u · u)(u ¯ · E(r, 0) + Fem (0) = α ρ(r, 0) 1 − (B.36) a c4 γ + 2 c

charge

˙ · r) u u u(u ¯ u − Iγ + 2 (1 − γ ) · × (u × E(r, 0)) dV + O(a) u γ 2 c2

136

Appendix B Force and Power at Arbitrary Velocity

as a approaches zero, where (1/a) in (B.36) denotes the 1/a terms. Inserting E(r, 0) from (C.1) into (B.36), noting that all odd terms integrate to zero, and extracting the 1/a terms, we ﬁnd

1 1 2¨ u γ 2¨ u ¯ · 3 + 2 u × u × 3 + α de de + O(a) Fem (0) = a 4π0 3c c 3c charge

e2 1 γ ¯ ¨ ¨ + = α · u + u × (u × u ) + O(a) a 6π0 c3 c2

(B.37)

where we have let de = ρ(r, 0)dV and performed the double integration of the constant integrand over the charge. With ¨ )u (u · u ¨ = γu ¯ · u ¨ + (1 − γ ) (B.38a) α 2 u and −γ u2 γ γ ¨) = ¨ )u ¨ + 2 (u · u u × (u × u (B.38b) u 2 2 c c c (B.37) can be written as

¨ ¨ )u e2 u 1 1 (u · u + (B.39a) Fem (0) = + 1− a 6π0 c3 γ γ u2

e2 1 ¨ + u ¨ ⊥ /γ + O(a) . + u (B.39b) 3 a 6π0 c ¨ and u ¨ ⊥ , in the proper K frame can The derivatives of the acceleration, u be expressed in terms of the velocity and its derivatives in the arbitrary K frame by means of the relativistic transformations (A.22). Using these transformations (A.22) converts (B.39b) to

e2 γ 2 1 3γ 2 ¨ + u + (u · u˙ )u˙ Fem (t ) = a 6π0 c3 c2 γ 2 3γ 2 2 ¨ + 2 (u · u˙ ) u + O(a) (B.40) + 2 u ·u c c or

Fem (0) =

where t has replaced t = 0 in (B.39) since the time t = 0 could be any instant of time t . The order unity term in (B.40) is the radiation reaction part of the self electromagnetic force. Combining the 1/a part of the self electromagnetic force in (B.19) with the radiation reaction part in (B.40) produces the total electromagnetic self force to order a in an arbitrary K inertial reference frame d e2 e2 γ 2 3γ 2 ¨ Fem (t ) = − u u ) + + (u · u˙ )u˙ (γ 6π0 ac2 dt 6π0 c3 c2 γ 2 3γ 2 2 ¨ + 2 (u · u˙ ) u + O(a) . (B.41) + 2 u ·u c c

B.2

Radiation Reaction of Self Electromagnetic Force and Power

137

B.2.2 Evaluation of the Radiation Reaction Power To determine the O(1) terms of the self electromagnetic power in an arbitrary K frame, begin with the charge integral in equation (3.2) at an arbitrary instant of time t = 0 Pem (0) = ρ (r , 0)u (r , 0) · E (r , 0) dV . (B.42) charge

Applying the same procedure to (B.42) as we applied to (B.24) in the previous section yields instead of (B.39b)

e2 1 ¨ . Pem (0) = + u · u (B.43) a 6π0 c3 ¨ from (A.22a) into (B.43), rearranging the expression, and Substituting u replacing the arbitrary time t = 0 with t , results in the radiation reaction power in the arbitrary K frame

e2 γ 4 1 3γ 2 2 ¨ ˙ + u + O(a) . (B.44) (t) = · u + (u · u ) Pem a 6π0 c3 c2 The 1/a part of the self electromagnetic power in (B.23) combines with (B.44) to give the total self electromagnetic power to order a in an arbitrary K inertial reference frame

e2 γ 4 e2 d 1 3γ 2 2 ¨ + 2 (u · u˙ ) + O(a). Pem (t) = − γ − + u ·u 6π0 a dt 4γ 6π0 c3 c (B.45) This completes the derivation of the self electromagnetic force and power to order a of Lorentz’s model of the electron, that is, a total charge e uniformly distributed on a spherical insulator of radius a moving without rotation with arbitrary center velocity u . To my knowledge, it is the ﬁrst rigorous derivation of these results for arbitrary velocity since Schott’s [16] rigorous, yet extraordinarily lengthy derivation from the Li´enard-Wiechert potentials; see Chapter 3 of the main text.

C Electric and Magnetic Fields in a Spherical Shell of Charge

Consider the Lorentz model of the electron as a total charge e uniformly distributed within a thin, nonrotating, spherical shell of inner radius a and thickness δ (see Fig. 4.1 of the main text). In a proper inertial reference frame at rest instantaneously with the charge distribution, the velocity u(r, t) will be zero but the acceleration and higher time derivatives of velocity are, in ˙ t), u ¨ (r, t), . . . ]. general, nonzero functions of space and time [u(r, In equation (A.10) of Appendix A the electric ﬁeld produced by this accelerating charge in its proper frame was found to be ! % ˆ " r · u˙ 1 1 R ˆ · u) ˆ + u˙ ˙ E(r, t) = + − 1 ( R R 4π0 R2 2c2 R c2 charge

' " 3(R ˆ · u) ˆ ! ˙ ˙ 3R u 2¨ u 2 2 ˆ · u) ˙ − |u| ˙ + 4 (R + + 3 + O(R) de , 8c 4c4 3c

u = 0 (C.1)

¨ in (C.1) refer to the time derivatives of the center velocity where u˙ and u of the charged sphere at time t. The position of the charge element de is designated by r (t) and the vector R is deﬁned as r − r (t). We can ﬁnd the magnetic ﬁeld B(r, t) from the simple relationship between the electric and magnetic ﬁelds of a moving point charge [13]. Letting de be the moving point charge, and dE(r, t) and dB(r, t) be the electric and magnetic ﬁelds of this point charge, we have ˆ (t ) × dE(r, t)/c dB(r, t) = R

(C.2)

where dE(r, t) is the integrand of (C.1) and R (t ) is deﬁned as r − r (t ), the diﬀerence vector between the position r of the observation point and the position r (t ) of the element of charge de at the retarded time t = t − R /c. ˆ (t ) in a power series about t and making use of (A.8) gives Expanding R

140

Appendix C Electric and Magnetic Fields in Spherical Shell

( ! " ˆ · u) ˙ 2 R r · u˙ (R 2ˆ ˆ ˆ ˆ ˆ ˙ ˙ − 1 ( R · u) R − u − R R (t ) = R − 2 R 2c c2 8c4 ) ( ) ˆ · u) ˆ ·u ˙ u˙ ˙ 2 ¨) ¨ (R |u| (R u + R2 + 4 + + 3 + O(R3 ). 3 8c 6c 4c4 6c

(C.3)

ˆ (t ) from (C.3) and dE(r, t) from the integrand of (C.1) into Substituting R (C.2), one ﬁnds that most of the terms cancel leaving merely ( ) ˆ ¨ R(t) ×u dB(r, t) = + O(R) de (C.4) 8π0 c4 or 1 B(r, t) = 4π0

(

) ˆ ¨ R(t) ×u + O(R) de , 2c4

u=0

(C.5)

charge

for the magnetic ﬁeld in the proper frame. Equations (C.1) and (C.5) can be integrated in closed form for a uniformly distributed spherical shell of charge with inner radius a and small thickness δ. In particular, the expressions for the ﬁelds within the thin shell simplify to

¯I|u| ˙ 2 r−a 2¨ u 4 2u˙ e ˆ ˆ ˙ ˙ + O(a) (C.6) + 3 + 4 r · uu − r− E(r, t) = 4π0 δa2 3ac2 3c 5c 3 B(r, t) =

e ˆr × u ¨ + O(a) , 12π0 c4

(C.7)

u = 0 , (a ≤ r ≤ a + δ) . The electric ﬁeld in (C.6) agrees with the results of Page and Adams [70, secs 56–57] except for the 4/5 term in (C.6), which is missing in their work, because they do not take into account the variation (A.8) in acceleration of the charge ¨ with position around the shell. Also Page and Adams do not include the u term in the magnetic ﬁeld of (C.7).

D Derivation of the Linear Terms for the Self Electromagnetic Force

Begin the derivation with the expression (A.2) for the electric ﬁeld produced by the moving element of charge de in the shell of charge. Since we want to evaluate this expression (A.2) in a proper reference frame (u(r, t) = 0) dis˙ u ¨ , . . . , we see, with the help of the expansion carding all nonlinear terms in u, ˆ (t ), and (A.8) and (A.13) for u(r ˙ , t ) and u(r , t ), that (A.2) can (C.3) for R be simpliﬁed immediately to ( ˆ × (R ˆ × u(t ˙ )) de R dE(r, t) = 4π0 Rc2 ) ˆ − u(t )/c R + nonlinear terms (D.1) + ˆ · u(t )/c)3 R2 (1 − R where, of course, R is a function of the retarded time t = t − R /c. Inserting the expansion (

ˆ · u(t ) R 1− c

)−3 =1+

ˆ · u(t ) 3R + nonlinear terms c

(D.2)

into (D.1) gives de dE(r, t) = 4π0

(

ˆ R ˆ · u(t ˙ ) u(t ) ˙ )) − u(t R( − 2 Rc2 R c ) ˆ (t ) ˆ R ˆ · u(t )) R 3R( + + nonlinear terms. + R2 c R2

(D.3)

Now R (t ) = R(t) −

˙ u(t) 2

R (t ) c

2 +

¨ (t) u 6

R (t ) c

3 + nonlinear terms (D.4)

142

Appendix D Derivation of the Linear Terms

or with the insertion of the expansion R · u˙ + ··· R (t ) = R 1 − 2c2

(D.5)

(D.4) becomes ˙ u(t) R (t ) = R − 2

that is Similarly,

R c

2

¨ (t) u + 6

R c

3 + · · · + nonlinear terms

R (t ) = R(t − R/c) + nonlinear terms.

(D.7)

u(t ) = u(t − R/c) + nonlinear terms

(D.8a)

and

(D.6)

˙ − R/c) + nonlinear terms ˙ ) = u(t u(t

(D.8b)

R (t ) = R(t − R/c) + nonlinear terms

(D.8c)

or 3 ¨ R 2 ... R R R R ˙ − + · · · + nonlinear terms. (D.8d) R (t ) = R − R + c 2 c 6 c

With

R dR d ˆ ·u=0 · =R R˙ = (R · R)1/2 = dt R dt ˆ · u˙ ¨=R R ... ˆ ·u ¨ + nonlinear terms R=R

(D.9a) (D.9b) (D.9c)

etc., inserted into (D.8d), R (t ) becomes R (t ) = R +

3 ˆ · u˙ R 2 ˆ ·u ¨ R R R +− + · · · + nonlinear terms. (D.10) 2 c 6 c

The vector R(t − R/c) can also be expanded in the form R(t − R/c) = R +

u˙ 2

R c

2 −

¨ (t) u 6

R c

3 + · · · + nonlinear terms (D.11)

which combines with (D.10) and (D.7) to give . ( /) 3 ˆ ·u ˆ · u˙ R 2 R ¨ R R (t ) R(t − R/c) R 3 R = = 3 1− − + ··· R3 (t ) R3 (t ) R R 2 c 6 c (

) 3 2 ¨ (t) R 1 u˙ R u + 3 − + · · · + nonlinear terms. (D.12) R 2 c 6 c

Appendix D Derivation of the Linear Terms

143

When we substitute (D.8a), (D.8b) and (D.12) into (D.3), integrate over de , then multiply by de = ρdV and integrate over de to get the total self electromagnetic force, we are left with integrals of the form [17] 2 x m 2m+1 m 2 Rm de de = 3 R de de = a e , R2 m+2 charge

charge

m = −1, 0, 1, 2, . . . . We see from (D.13) applied to (D.12) that R (t ) de de = 0 + nonlinear terms. R3 (t )

(D.13)

(D.14a)

charge

Similarly, from (D.13) applied to the u(t ) part of (D.3) u(t ) ! ˆ ˆ ¯" · 3RR − I de de = 0 + nonlinear terms R2

(D.14b)

charge

˙ ) part of (D.3) and from (D.13) applied to the u(t ˙ ) ! ˆ ˆ ¯" ˙ − R/c) u(t u(t 2 · RR − I de de = − de de R 3 R charge

charge

+ nonlinear terms.

(D.14c)

Thus, integrating (D.3) over de and de and using (D.14) shows that the exact expression for the total self electromagnetic force on the charge can be written simply as ˙ − R/c) u(t 1 de de dE(r, t)de = − Fem (t) = 6π0 c2 R charge

charge

+ nonlinear terms.

(D.15)

˙ − R/c) can be expanded in the power series Since u(t

n ∞ 1 dn+1 u(t) −R ˙ − R/c) = u(t n! dtn+1 c n=0

(D.16)

substituting (D.16) into (D.15) and applying the integrals (D.13) yields

n+1 ∞ dn+1 u(t) −2a 1 e2 Fem (t) = + nonlinear terms 12π0 a2 c n=0 c (n + 1)! dtn+1 (D.17) or

144

Appendix D Derivation of the Linear Terms

Fem (t) =

e2 u(t − 2a/c) + nonlinear terms, u(t) = 0 12π0 a2 c

(D.18)

or for small velocity u2 e2 [u(t − 2a/c) − u(t)] + nonlinear terms, 1. 12π0 a2 c c2 (D.19) The result (D.18) was stated without proof by Page [17]. It can also be obtained from the ﬁrst series of a general expression for the self electromagnetic force, on a nonrelativistically rigid charged sphere, that was derived by Schott [64]. The linear part of the self electromagnetic force (D.19) is the same for both relativistically and nonrelativistically rigid spheres. Fem (t) =

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Index

Abraham, 1–3, 5, 11, 13, 14, 18, 27, 38, 42, 43, 66, 78, 79, 94 nonrelativistically rigid model, 18, 42, 43, 75, 113, 144 Adams, 140 analyticity, 12, 73, 76, 77, 79, 85, 89 external force, 77, 85, 86, 117 velocity, 77 antenna reactance, 62 resistance, 62 asymptotic condition on acceleration, 65, 69 on velocity, 70, 84, 88 bare mass, 13, 28, 30, 35, 39, 43, 75 corrected term, 80 Bauer, 75 Baylis, 81 binding force, 1–6, 23–37, 39, 45–50, 52–55, 58, 128 Bohr, 43 Bucherer, 43 Buchwald, 11 Caldirola relativistic generalization of ﬁnite diﬀerence equation of motion, 113 center velocity, 11, 35, 123 charged insulating sphere, 23, 24, 26, 27, 35 classical radius of electron, 13, 68 conservation of momentum-energy, 93, 96, 117

Coulomb ﬁeld, 89 Cunningham, 43 D¨ urr, 75 Dirac, 6, 42, 117 delta function, 50 Einstein, 11, 13, 24 mass-energy relation, 1, 3, 5, 6, 13, 33, 35, 39, 42, 45, 47, 76 electromagnetic ﬁelds in spherical shell of charge, 139 electric ﬁeld, 140 magnetic ﬁeld, 140 electromagnetic mass, 13, 14, 31, 35, 37, 39 electrostatic energy of formation, 13, 14, 39, 40 electrostatic mass, 13, 35, 38–40 energy bare-mass, 49, 53 binding, 51, 53 electromagnetic, 47 total supplied by external force, 59 equation of motion corrected, 80–82 formal solution to, 82 four-vector notation, 81 rectilinear, 84 ﬁnite diﬀerence, 112, 113 relativistic generalization, 114 ﬂawed expression for, 80 force, 39, 62 four-vector notation, 59

150

Index

general, 102 renormalized, 103, 115 power, 39, 62 proper-frame, 99 force, 19–21, 76 power, 19–21 renormalized, 100 rectilinear, 100 renormalized, 102 solutions to, 67 general, 72 peculiarities of, 70 power series, 98 rectilinear, 68, 70 uniform electric ﬁeld, 91 uniform magnetic ﬁeld, 108 external force, 12, 28 electromagnetic, 104 per unit charge, 35 external power, 14 per unit charge, 35 Fermi, 54 force gravitational, 39, 40 short-range, 39, 40 total on charge, 36, 37 force-power, 31, 40 bare-mass, 45, 46, 48 binding, 45, 46, 48 electromagnetic, 23, 45, 48 fundamental equation of motion for charged particle, 8, 38, 80 gravitational and other attractive formation energies, 39, 40, 116 Herglotz, 75, 114 Hertz, 94 Huschilt, 81 hyperbolic motion, 63, 92, 108 between parallel plates, 64 constant external force, 63 relativistically uniform acceleration, 63 insulator mass, 26, 29, 30, 35, 38–40, 42, 75 internal binding energy, 25, 26

internal binding force, 23, 24, 26, 28, 31, 33, 35 electric polarization producing, 24, 32 eﬀective molecular polarizability, 32, 33 electric susceptibility, 33 homogeneous, 29, 30 inhomogeneous, 30 per unit charge, 35 internal binding power, 26, 31 per unit charge, 35 jumps in velocity and acceleration across transition intervals, 88, 90, 91, 93–97, 115, 117 Kaufmann, 38, 42, 43 Landau-Lifshitz, 9, 104 approximation to equation of motion, 9, 98, 105, 108, 111, 112 for charge in uniform magnetic ﬁeld, 108 three-vector part, 106 Laue, von, 15 Li´enard-Wiechert electric ﬁelds, 4 potentials, 3, 18, 19, 137 Lorentz, 1–3, 5, 11–15, 17–19, 24, 27, 28, 38, 42, 43, 50, 53, 64, 121 bare mass, 42 force, 6, 11, 17, 25, 35, 36, 47, 58, 80 proper-frame, 122 force density, 3 model of electron, 1–4, 6, 11–14, 18, 19, 24, 27, 32, 35, 42, 94, 114, 137 power, 6, 17, 36, 58 power density, 3 relativistically rigid model, 2, 4, 11, 12, 17, 23, 24, 27, 31, 36, 42, 43, 113, 114, 144 Lorentz-Abraham, 17, 20 equation of motion, 1, 5, 7, 11, 40, 79 derivation of force, 17–21 derivation of power, 17–21 force, 2–5, 11–13, 23, 26, 31, 33, 36, 45 four-vector notation, 15 power, 2, 4, 5, 13–15, 23, 26, 31, 33, 37, 45

Index Lorentz-Abraham-Dirac equation of motion, 5, 7–9, 15, 79, 100, 103, 105, 110, 111, 115, 117 corrected, 115, 116 high acceleration catastrophe, 117 magnetars, 111 material mass, 13, 42 Maxwell equations, 11, 14, 17, 24, 25, 35, 36, 47, 58, 80, 94, 108, 131 causal solutions to, 74 Maxwell stress tensor, 129, 130 Maxwell-Lorentz equations, 1, 2, 5, 117 measured mass, 13, 38–40 metric tensor, 51 momentum bare-mass, 49 binding, 51 electromagnetic, 47 total supplied by external force, 59 momentum-energy, 31, 40 angular, 41 bare-mass, 45, 46 binding, 45, 46 electromagnetic, 20, 23, 45, 46 redeﬁnition, 45, 54 extra in equation of motion, 42 irreversible, 60 reversible, 60 necessary condition on magnetic ﬁeld, 111 Neumann, 43 Newton second law of motion, 1, 2, 4–7, 13, 30, 35, 36, 38, 42, 43, 45, 47, 48, 60, 64, 68, 74, 76, 80, 92 extra momentum-energy in, 40 nonrelativistic, 42 Newtonian acceleration, 5, 9, 93, 116 Nodvik, 12 rotating charged sphere, 2, 12 Page, 19, 112, 140, 144 Panofsky and Phillips, 15 parallel-plate capacitor, 64, 92, 93, 96 Pauli, 15, 43 Pearle, 114 Plass, 110 Poincar´e, 1–4, 23, 24, 26, 30, 31

151

binding force, 1, 3–5, 23, 24, 26, 27, 31, 57 postulate of relativity, 24 power total delivered to charge, 36, 37 power series, 21, 41, 74, 89 power series solution general equation of motion, 102, 103 renormalized, 103, 104 proper-frame equation of motion, 99 renormalized, 100 rectilinear equation of motion, 100, 101 renormalized, 102, 108 Poynting vector, 62, 130 pre-acceleration, 12, 21, 41, 64, 70, 72, 73, 78, 79, 88, 90, 101, 114 cause of, 73, 76 elimination of, 73, 74, 79, 81, 87, 112 pre-deceleration, 64, 73, 90 elimination of, 81, 87 quantum mechanics, 33, 105, 112, 117 radiated momentum-energy, 60, 61, 65, 66, 94, 102 during transition intervals, 94, 95, 97, 115, 116 radiation reaction, 13, 15, 17, 74, 75 four-vector notation, 15 irreversible part, 41, 62 reversible part, 41, 62 radiationless motion, 114 rectilinear motion, 63–65, 67, 89 oscillating charge, 61 renormalization, 15, 89, 96, 115, 117 retarded time, 76, 122 Richardson, 43 Rohrlich, 49, 55, 69, 105, 112 runaway motion, 64, 74, 88, 92, 101, 114 dumbbell problem, 75 Schott, 1–3, 18, 19, 21, 43, 67, 79, 137 acceleration momentum-energy, 7, 61–65, 71, 94, 95 four-vector notation, 15 double inﬁnite series for self electromagnetic force, 113, 144 runaway motion, 65, 69 spinning sphere, 12

152

Index

Schwinger, 57, 110 stress-momentum-energy tensor, 6, 40, 54, 57 self electromagnetic force, 13, 17, 23, 28, 76, 129 corrected expression, 78, 80 derivation of linear terms, 141, 144 evaluation of 1/a term, 130, 132 evaluation of radiation reaction term, 134, 136 per unit charge, 35 proper-frame, 123 derivation, 121 relativistic transformation, 125, 126 self electromagnetic power, 14, 17, 23, 129 evaluation of 1/a term, 133 evaluation of radiation reaction term, 137 noncovariance, 127 per unit charge, 35 small-velocity derivation, 124, 125 relativistic transformation, 126 self force, 17, 20, 41, 58 self power, 17, 20, 58 Shen, 110 Sommerfeld, 79 Spohn, 105 charge in uniform magnetic ﬁeld, 9, 108

step function, 78 stress-momentum-energy tensor bare-mass, 49 binding, 51 electromagnetic, 20, 48 Schwinger, 6, 40, 54, 57 total, 45, 48, 54 successive substitutions, 98–100, 102, 103 suﬃcient conditions on external force, 12, 99, 100, 102–105, 116 on magnetic ﬁeld, 110, 111 on velocity, 12, 74, 115 Taylor series, 7, 76, 77, 101 Thomson, 11, 19, 43 transition force, 21, 41, 78, 81, 83, 87, 112 corrected Lorentz-Abraham-Dirac equation of motion, 115 four-vector notation, 81 rectilinear motion, 89 transition interval, 41, 64, 77, 78, 80, 87 uniform electric ﬁeld, 64, 91 uniform magnetic ﬁeld, 108, 110–112 Valentini, 79 Wildermuth, 75, 114

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